Hierarchy construction and non-Abelian families of generic topological orders
Tian Lan Xiao-Gang Wen







 
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 
Department of Physics and Astronomy,
  University of Waterloo, Waterloo, Ontario N2L 3G1, Canada


Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA

 
We generalize the hierarchy construction to generic 2+1D topological orders
(which can be non-Abelian) by condensing Abelian anyons in one topological
order to construct a new one.  We show that such construction is reversible and
leads to a new equivalence relation between
topological orders. We refer to the corresponding equivalent class (the orbit of
the hierarchy construction) as “the non-Abelian family”.  Each non-Abelian
family has one or a few root topological orders with the smallest number of
anyon types.  All the Abelian topological orders belong to the trivial
non-Abelian family whose root is the trivial topological order.  We show that
Abelian anyons in root topological orders must be bosons or
fermions with trivial mutual statistics between them.  The classification of
topological orders is then greatly simplified, by focusing on the roots of each
family: those roots are given by non-Abelian modular extensions of
representation categories of Abelian groups.





Hierarchy construction and non-Abelian families of generic topological orders
    Xiao-Gang Wen
    
=============================================================================








Introduction:
The ultimate dream of classifying objects in nature may be creating a “table”
for them. A classic example of such classification result is the “Periodic
Table” for chemical elements. As for the topological ordered<cit.>
phases of matter, which draws more and more research interests recently, we are
already able to create some “tables” for
them<cit.>, via
the theory of 
categories.  However, efforts are needed to further understand the tables, for
example, to reveal some “periodic” structures in the table.

In the Periodic Table, elements are divided into several “families” (the
columns of the table), and those in the same family have similar chemical
properties. The underlying reason for this is that elements in the same family
have similar outer electron structures, and only differ by “noble gas cores”.
The last family consists of noble gas elements, which are chemically “inert”,
as they have no outer electrons besides the noble gas cores. Thus the
“family” can be considered as the equivalent class up to the “inert” noble
gas elements.

When it comes to topological orders, we also have “inert” ones: the Abelian
topological orders are “inert”, for example, in the application of
topological quantum computation<cit.>. Abelian anyons can not
support non-local topological degeneracy, which is an essential difference from
non-Abelian anyons. Is it possible to define equivalent classes for topological
orders, which are up to Abelian topological orders? In this letter, we use the
hierarchy construction to establish such equivalent classes, which we will call
the “non-Abelian families”.  The hierarchy construction is well known in the
study of Abelian fractional quantum Hall (FQH)
states<cit.>.  In this letter we generalize it to
arbitrary (potentially non-Abelian) topological orders. 

We show that the generalized hierarchy construction is reversible. Thus, we
can say that two topological orders belong to the same “non-Abelian family”
if they are related by the hierarchy construction. 


Each non-Abelian family has special “root” topological orders (see Table
<ref>), with the following properties:

  
  * Root states have the smallest rank (number of anyon types) among
    the non-Abelian family.
  
  * Abelian anyons in a root state are all bosons or fermions, and have
    trivial mutual statistics with each other.

Since any topological order in the same non-Abelian family can be reconstructed
from a root state, our work simplifies the classification of generic
topological orders to the classification of root states.


Our calculation is based on quantitative characterizations of topological
orders. One way to do so is to use the S,T modular matrices obtained
from the non-Abelian geometric phases of degenerate ground states on torus
<cit.>. We will show,  starting from a topological order described
by S,T,  how to obtain another topological order described by new S',T' via
a condensation of Abelian anyons. (For a less general approach based on wave
functions, see BS07113204.) The calculation uses the theory of fusion and
braiding of quasiparticles (which will be called anyons) in topological
order. Such a theory is the so called “unitary modular tensor category (UMTC) theory”
(for a review and much more details on UMTC, see W150605768).

A UMTC  is simply a set of anyons (two anyons connected by a local operator
are regarded as the same type), plus data to describe their fusion and
braiding.  Like the fusion of two spin-1 particles give rise to a “direct
sum” of spin-0,1,2 particles: 1 ⊗ 1 =0⊕ 1 ⊕ 2, the fusion
of two  anyons i and j in general gives rise to a “direct sum” of several
other anyons: i⊗ j = ⊕_k N^ij_k k.  So the fusion of anyons
is quantitatively described by a rank-3 integer tensor N^ij_k.  From
N^ij_k, we can determine the internal degrees of freedom of anyons, which
is the so called quantum dimension. For example, the quantum dimension
of a spin-S particle is d=2S+1.  For an anyon i, its quantum dimension
d_i, which can be non-integer, is the largest eigenvalue of matrix N_i with
(N_i)_kj=N^ij_k.

After knowing the fusion, the braiding of anyons can be fully determined by the
fractional part of their angular momentum L^z: s_i =mod(L^z_i,1).
s_i is called the topological spin (or simply spin) of the anyon i.
The last piece of data to characterize topological orders is the chiral
central charge c, which is the number of right-moving edge modes minus the
number of left-moving edge modes.  

It turns out that two sets of data (S,T) and (N^ij_k,s_i) can fully
determine each other:

    T_ij   =^ 2π s_i_ij,
       
     S_ij   =∑_k  ^ 2π (s_i+s_j-s_k)N^ij_k d_k/D,
    
    ^ 2π s_i   =T_ii, 
       
    N^ij_k    =∑_l S_li S_ljS_lk/ S_l1 .

where D=√(∑ d_i^2) is the total quantum dimension.

1.25 


 

Hierarchy construction in generic topological orders: Let us consider
an Abelian anyon condensation in a generic topological order, described by a
UMTC .  (Such a condensation in an Abelian topological order is
discussed in Appendix <ref>.) The anyons in  are labeled by
i,j,k,⋯.  Let a_c be an Abelian anyon in  with spin s_c.  We
condense a_c into the Laughlin state Ψ=∏(z_i-z_j)^m_c-2s_c,

where m_c= even and m_c-2s_c≠ 0.  [This is different from the
so called “anyon condensation” categorical approach where only bosons
condensing into the trivial state is considered.] The resulting topological
order is described by UMTC , determined by , a_c and m_c.  


To calculate , we note that the anyons in  are the anyons i in
 dressed with the vortices of the Laughlin state of a_c.  The vorticity
is given by m-t_i, where m is an integer, and 2π t_i is the mutual
statistics angle between anyon i and the condensing anyon a_c in the
original topological order , which can be extracted from the S matrix


^-2π t_i=S^_i a_c/S^_i, or -t_i = s_i + s_a_c -
s_i⊗ a_c.  Thus anyons in  are labeled by pairs I=(i,m).  We
like to ask what is the spin and fusion rules of I=(i,m)?  

The spin of (i,m) is given by the spin of i plus the spin of the m-t_i
flux in the Laughlin state:

    s_(i,m)    = s_i +1/2(m-t_i)^2 /m_c-2s_c

Fusing i with m-t_i flux and j with n-t_j flux gives us
i⊗ j with m-t_i + n-t_j flux:


    (i,m)   ⊗(j,n) ∼⊕_kN^ij_k(k,m-t_i+n-t_j+t_k),

where N^ij_k is the fusion coefficient in .
Since a_c with m_c-t_a_c=m_c-2s_c flux is condensed, fusing with
(a_c,m_c) anyon does not change the anyon type in .  So, we have
an equivalence relation:

    (i,m)    ∼ (i⊗ a_c, m-t_i+m_c-2s_c+t_i⊗ a_c),

The above three relations fully determine the topological order .<cit.>

It is important to fix a “gauge” for t_i, say by choosing t_i
∈ [0,1).  The same label (i,m) may label different anyons under different
“gauge” choices of t_i. Similarly, we have fixed a “gauge” for s_c that
fixed the meaning of m_c.  Note that t_a_c is automatically fixed when
s_c is fixed: t_a_c=2 s_c, while other t_i can be freely chosen. This
ensures that the equivalence relation (<ref>) is compatible with fusion
(<ref>), where (<ref>) is generated by fusing with the
trivial anyon (a_c,m_c).  The combinations m-t_i,
m_c-2s_c determine the final spins and fusion rules; they are
gauge-invariant quantities. Thus, if we change the gauge of t_i,s_c, i.e.,
modify them by some integers, m,m_c should be modified by the same integers to
ensure that the construction remains the same.

Below we will study the properties of  in detail. Let M_c=m_c-2s_c.
Applying the equivalence relation (<ref>) q times, we obtain

    (i,m)∼ (i⊗ a_c^⊗ q,m-t_i+qM_c+t_i⊗ a_c^⊗
      q).

Let q_c be the “period” of a_c, i.e., the smallest positive integer such
that a_c^⊗ q_c=. We see that

    (i,m)∼ (i,m+q_cM_c).

Thus, we can focus on the reduced range of m∈{0,1,2,⋯,q_c|M_c|-1}.
Let ||, || denote the rank of , respectively. Now within the
reduce range of m, we have q_c|M_c||| different labels, and we want to
show that the orbit generated by the equivalence relation (<ref>) all
have the same length, which is q_c. To see this, just note that for
0<q<q_c, either i≠ i⊗ a_c^⊗ q, or if i=i⊗
a_c^⊗ q, m≠ m-t_i+qM_c+t_i⊗ a_c^⊗ q=m+qM_c; in
other words, the labels (i,m) are all different within q_c steps. It
follows that the rank of  is ||=|M_c|||.  

Strictly speaking, anyons in  should one-to-one correspond to the
equivalent classes of (i,m). However, as the orbits have the same length, it
would be more convenient to use (i,m) directly (as we will see later, this
is the same as working in a pre-modular category ). For example, when we need to
sum over anyons in , we can instead do
∑_I∈→1/q_c∑_i∈∑_m=0^q_c|M_c|-1.
Now we are ready to calculate other quantities of the new topological order
. First, it is easy to see that the quantum dimensions remain the same
d_(i,m)=d_i.
The total quantum dimension is then

    D_^2=1/q_c∑_i∈∑_m=0^q_c|M_c|-1
      d_(i,m)^2=|M_c|D_^2.

The S matrix is

    S^_(i,m),(j,n)   =∑_k N^ij_k/D_ d_k ^2π
        [s_(i,m)+s_(j,n)-s_(k,m+n+t_k-t_i-t_j)]
       =1/√(|M_c|)S^_ij^-2π(m-t_i)(n-t_j)/M_c.

It is straightforward to check that S^_(i,m),(j,n) is unitary (with
respect to equivalent classes of (i,m)). Moreover, this formula for S can
recover the equivalence relation (<ref>) and fusion rules
(<ref>) via unitarity and Verlinde formula.

The new S^,T^ matrices (T^-matrix is determined by the spin of
anyons s^_(i,m) in (<ref>)), as well as S^,T^, should
both obey the modular relations STS=^2πc/8T^† S T^†,
from which we can extract the central charge of . The new
central charge is found to be (see Appendix <ref>)

    c^=c^+(M_c) .

Clearly, the one-step hierarchy construction described by (<ref>),
(<ref>), and (<ref>) is fully determined by an Abelian anyon a_c and
M_c, where M_c+2s_c is an even integer.  In Appendix <ref>,  we
discuss the above hierarchy construction more rigorously at the full
categorical level.



As an application, let us explain  the “eight-fold way” observed in the table
of topological orders<cit.>: whenever there is a
fermionic quasiparticle, the topological order has eight companions with the
same rank and quantum dimensions but different spins and central charges. If we
apply the one-step condensation with a_c being a fermion, and M_c=± 1, a
new topological order of the same rank is obtained.  [Physically this
amounts to condensing the fermionic quasiparticle into an integer quantum Hall
state. If we instead condense the fermionic quasiparticle into p± p
states we are able to obtain the “sixteen fold way”.  However, such
condensation is beyond the construction of this work.] The spins of the anyons
carrying fermion parity flux (having non-trivial mutual statistics with the
fermion a_c) are shifted by ±1/8, and the central charge is shifted by
± 1, while all the quantum dimensions remain the same.  If we repeat it
eight times, we will go back to the original state (up to an E_8 state), generating the
“eight fold way”.

 

Reverse construction and non-Abelian families:
The one-step condensation from  to  is always reversible. In ,
choosing a_c'=(,1), s_c'=1/2M_c, m_c'=0, M_c'=-1/M_c, and repeating
the construction, we will go back to .  


One may first perform the construction for a pre-modular  and then
reduce the resulting category to a modular category. Taking
(j,n)=a_c'=(,-1) in (<ref>) we find that the mutual
statistics between (i,m) and a_c'=(,1) is
t'_(i,m)=m-t_i/M_c.



Let (i,m,p),(j,n,q) label the anyons after the above one-step condensation;
the new S matrix is

    S_(i,m,p),(j,n,q)=S^_ij^-2π(m-t_i)(n-t_j)/M_c^-2π(p-t'_(i,m))(q-t_(j,n)')/M'_c
         =S^_ij^2π (t_i q+ t_j p -t_a_c pq)
      =S^_i⊗a̅_c^⊗ p,j⊗a̅_c^⊗ q,

which means that we can identify (i,m,p) with i⊗a̅_c^⊗ p
(a̅_c denotes the anti-particle of a_c).
It is easy to check that they have the same spin s_(i,m,p)=s_i⊗a̅_c^⊗ p. Therefore, i∼ (i⊗ a_c^⊗ p,m,p), ∀
  m,p, we have come back to the original state .
Therefore, generic hierarchy constructions are reversible, which defines an
equivalence relation between topological orders. We call the corresponding
equivalent classes the “non-Abelian families”.

Now we examine the important quantity M_c=m_c-2s_c which relates the ranks
before and after the one-step condensation, ||=|M_c|||. Since m_c is
a freely chosen even integer, when a_c is not a boson or fermion (s_c≠ 0
or 1/2  1), we can always make 0<|M_c|<1, which means that the rank is
reduced after one-step condensation. We then have the first important
conclusion: Each non-Abelian family have “root” topological orders with
the smallest rank; the Abelian anyons in the “root” states are all bosons or
fermions.

We can further show that the Abelian bosons or
fermions in the “root” states have trivial mutual statistics among them.
To see this, assuming that a,b are Abelian anyons in a root state. Since
the mutual statistics is given by DS_ab=exp[2π(s_a+s_b-s_a⊗
b)], and a,b,a⊗ b are all bosons or fermions, non-trivial mutual
statistics can only be DS_ab=-1. Now consider two cases: (1) one of a,b,
say a, is a fermion, then by condensing a (choosing a_c=a,
m_c=2, s_c=1/2, t_b=1/2), in the new topological order, the rank remains the same but
s_(b,0)=s_b+t_b^2/2M_c=s_b+1/8, which means (b,0) is an Abelian
anyon but neither a boson nor a fermion. By condensing (b,0) again we can
reduce the rank, which conflicts with the “root” state assumption. (2) a,b
are all bosons. Still we condense a with m_c=2,s_c=0,t_b=1/2. In the new
topological order the rank is doubled but
s_(b,0)=s_b+t_b^2/2M_c=1/16, which means further condensing (b,0)
with m_c'=0 the rank is reduced to 1/8, which is again, smaller than the
rank of the beginning root state, thus contradictory.

Therefore, in the root states, Abelian anyons are bosons or fermions with
trivial mutual statistics.


We also have a straightforward corollary: all Abelian topological orders
have the same unique root state, which is the trivial topological order.  In
other words, all Abelian topological orders are in the same trivial non-Abelian
family, which resembles the noble gas family in the Periodic Table. Thus, the
non-Abelian families are indeed equivalent classes up to Abelian topological
orders.

To easily determine if two states belong to the same non-Abelian family, 


it is very helpful to introduce some non-Abelian invariants. One is the fractional part of
the central charge.  Since the one-step condensation changes the central charge
by (M_c) (see (<ref>)), we know that central charges in the same
non-Abelian family can only differ by integers.




Another invariant is the quantum dimension. It is not hard to check that, in
the one-step condensation, the number of anyons with the same quantum dimension
is also multiplied by |M_c|. The third invariant is a bit involved. Note that
in the one-step condensation, if i has trivial mutual statistics with a_c,
t_i=0, then (i,0) in  have the same spin as i in  and the same
mutual statistics with (j,m),∀ m as that between i and j in .
Therefore, the centralizer of Abelian anyons, namely, the subset of anyons that
have trivial mutual statistics with all Abelian anyons (the anyons in red in
Table <ref>), is the same within one non-Abelian family. These facts
enable us to quickly tell that two states are not in the same non-Abelian
family.

 

Examples: Realizations of non-Abelian FQH states were first
proposed in Wnab,MR9162. One of them 

is <cit.>

    Ψ_ν=1({z_i})=[χ_2({z_i})]^2,

where χ_k({z_i}) is the many-fermion wave function with k filled
Landau levels.  The bulk effective theory is the
SU(2)_-2^f Chern-Simons (CS) theory with 3 types of anyons and the edge has 

c=5/2 (see Appendix <ref>). So the state is one of the root state N_c=3_5/2 in Table
<ref>.  Another bosonic non-abelian FQH liquid at ν=1 is
<cit.>

    Ψ_ν=1 = (1/z_1-z_21/z_2-z_3⋯)∏ (z_i-z_j),
 
whose edge has a  chiral central charge c=3/2.  It is the state described by
N_c=3_3/2 which belong to the same non-Abelian family as the 3_5/2 state above.  The experimentally realized ν=5/2 FQH state is likely to
belong to this non-Abelian family <cit.>.  

A more interesting non-Abelian state (which can perform universal topological quantum computation <cit.>) is

    Ψ_ν=3/2({z_i})=[χ_3({z_i})]^2,

whose edge has a  chiral central charge  c=21/5. The bulk effective
theory is the SU(2)_-3^f CS theory with 4 types of anyons <cit.>.
So the state is N_c=4_21/5, which belongs to the same non-Abelian
family as the state 2_26/5 in Table <ref> (see Appendix
<ref>, which contains more examples of non-Abelian states and
non-Abelian families).




















We like to remark that the topological orders studied in this paper do not
require and do not have any symmetry. However, some c=0 topological orders
with a Z_2 automorphism i→ i' that changes the sign of spins
s_i =-s_i' can be realized by
time-reversal symmetric states <cit.>.

 


Conclusion and Outlook:
In this letter we introduced the hierarchy construction in generic topological
orders, which established a new equivalence relation: Two topological orders
related by the hierarchy construction belong to the same “non-Abelian
family”. This reveals intriguing new structures in the classification of
topological orders.



Non-Abelian families are equivalent classes up to Abelian topological orders.
Topological orders in the same non-Abelian family share some properties, such
as quantum dimensions and the fractional part of central charges.

In particular we studied the “root” states, the states in a non-Abelian
family with the smallest rank.  Other states can be constructed from the root
states via the hierarchy construction.  Thus, classifying all topological
orders is the same as classifying all root states, namely, all states such that
their Abelian anyons have trivial mutual statistics. In other words, we can try
to generate all possible topological orders by constructing all the root
states, which can be obtained by starting with an Abelian group G,
extending its representation category (G) or (G^f) to a modular
category<cit.> while requiring all the extra anyons
being non-Abelian (which is referred to as a non-Abelian modular extension).
This is a promising future problem and may be an efficient way to produce
tables of topological orders. 

Although in this letter we focused on bosonic topological orders with no
symmetry (described by modular categories), the construction also applies to
bosonic/fermionic topological orders with any symmetry (described by certain
pre-modular categories)<cit.>. The same argument goes
for non-Abelian families and root states with symmetries.  










TL thanks Zhenghan Wang for helpful discussions. This research was supported by NSF Grant No.  DMR-1506475 and NSFC 11274192.

 





§ HIERARCHY CONSTRUCTION IN ABELIAN TOPOLOGICAL ORDERS



In this section, we will discuss hierarchy construction, Abelian anyon
condensation, in Abelian topological orders in a very general setting.  This
motivates the similar construction for generic non-Abelian states discussed in
the main text.

Consider a bosonic Abelian topological order, which can always be described by
an even K-matrix K_0 of dimension .  Anyons are labeled by
-dimensional integer vectors ľ_0.  Two
integer vectors ľ_0 and ľ'_0 are equivalent (describe the same
type of topological excitation) if they are related by

    ľ'_0 = ľ_0 + K_0 ǩ,

where ǩ is an arbitrary integer vector.  The mutual statistical angle between two
anyons, ľ_0 and ǩ_0, is given by

    þ_ľ_0,ǩ_0 =2πǩ_0^T K_0^-1ľ_0.

The spin of the anyon ľ_0 is given by

    s_ľ_0 = 1/2ľ_0^T K_0^-1ľ_0.


In the hierarchy construction of a new topological order from an old one, a
basic step is to condense Abelian anyons into a Laughlin-like
state. Let us construct a new topological order from the K_0
topological order by assuming Abelian anyons labeled by ľ_c condense.
Here we treat the anyon as a bound state between a boson and flux.  We then
smear the flux such that it behaves like an additional uniform magnetic field,
and condense the boson into ν=1/m_c Laughlin state (where m_c= even).
The resulting new topological order is described by the (+1)-dimensional
K-matrix

    K_1=  K_0    ľ_c 
    ľ_c^T      m_c


In the following, we are going to show that, to describe the result of the ľ_c anyon condensation, we do not need to know K_0 directly.  We only need
to know the spin of the condensing particle ľ_c

    s_c=1/2ľ_c^T K_0^-1ľ_c,

and the mutual statistics 

    þ_ľ_0,ľ_c≡ 2π t_ľ_0,    t_ľ_0 = ľ_c^T K_0^-1ľ_0

between ľ_0 and  ľ_c.
A calculation:

    K_0    ľ_c 
    ľ_c^T      m_c 
     K'_0    ľ_c' 
    ľ_c^' T      m_c' 
       =
      K_0 K'_0 + ľ_c ľ_c^' T    K_0 ľ_c' +m_c' ľ_c
    ľ_c^T K'_0 + m_c ľ_c^' T     m_c m_c' + ľ_c^T ľ_c' 
    
        = I


    ľ_c'    = -m_c' K_0^-1ľ_c
    
    
    m_cm_c'    =1 -  ľ_c^T ľ_c' = 1 + m_c' ľ_c^T K_0^-1ľ_c

 

    m_c'     = 1/ m_c - ľ_c^T K_0^-1ľ_c =  1/ m_c - 2s_c
    ľ_c'    = - K_0^-1ľ_c/ m_c - 2s_c
    
     K_0'     = K_0^-1 (I-ľ_c ľ_c^' T) = K_0^-1 + K_0^-1ľ_c ľ_c^T K_0^-1/ m_c - 2s_c

to be removed


First, we find that, as long as m_c-2s_c≠ 0, K_1 is invertible with

    K_1^-1 = 
    
     K_0^-1 + K_0^-1ľ_c ľ_c^T K_0^-1/ m_c - 2s_c    - K_0^-1ľ_c/ m_c - 2s_c
    
    - ľ_c^T K_0^-1/ m_c - 2s_c   1/ m_c - 2s_c

The anyons in the new K_1 topological order are labeled by
+1-dimensional integer vector ľ^T = (ľ_0^T, m).  The spin of ľ is

    s_ľ   = 1/2ľ^T K_1^-1ľ
     = 1/2(2s_0 + m^2+t_ľ_0^2 -2 m t_ľ_0/m_c-2s_c) 
    
       = s_ľ_0 +1/2(m-t_ľ_0)^2 /m_c-2s_c

The vectors 
ľ^T = (ľ_0^T, m)
and
ľ^' T = (ľ_0^' T, m')
are equivalent if they are related by

    ľ_0^' - ľ_0 = K_0 ǩ_0 + k ľ_c,   m' - m = ľ_c^T ·ǩ_0+m_c k,

for any -dimensional integer vector ǩ_0 and integer k. 
To avoid the gauge ambiguity, for the
integer vectors ľ_0, we pick a representative for each equivalent class
(by (<ref>), fixing the gauge).
Taking k=1 and appropriate ǩ_0 such that ľ_0' and ľ_0 are
the pre-fixed representatives, we see that 

    (ľ_0^T, m) ∼ (ľ'_0^T∼ľ_0^T+ľ_c^T, m+t_ľ'_0-t_ľ_0+m_c-2s_c).



We also want to express the fusion in the new state in terms of the pre-fixed
representatives ľ_1,ľ_2,ľ_3. Assuming that (ľ_3^T,m_3)∼ (ľ_1^T+ľ_2^T,m_1+m_2), and taking k=0 and appropriate ǩ_0 in
(<ref>) (the cases of non-zero k can be generated via
(<ref>)), we find that

    (ľ_1^T,m_1)+(ľ_2^T,m_2)
    
       ∼ (ľ_3^T∼ľ_1^T+ľ_2^T,m_3=m_1+m_2+t_ľ_3-t_ľ_1-t_ľ_2).


We can easily calculate the determinant of K_1 whose absolute value is the
rank of the new state:

    (K_1)   = K_0    ľ_c 
    ľ_c^T      m_c 
     = (K_0)(m_c-ľ_c^T K_0^-1ľ_c)
         
       =(m_c-2s_c)(K_0)

Let M_c=m_c-2s_c. It is an important gauge invariant quantity relating the
ranks of the two states. If we perform the condensation with a different anyon ľ_c'
and a different even integer m_c', but make sure that ľ_c'∼ľ_c
and M_c'=m_c'-2s_c'=m_c-2s_c=M_c, the new topological order will be the same.

It is worth mentioning that such construction is reversible: for the K_1
state, take ľ_c'^T=(0̌^T,1),m_c'=0, and repeat the construction:

    K_2= K_0   ľ_c     0
    ľ_c^T    m_c   1
    
      0   1   0∼ K_0    0     0
    
      0    0   1
    
      0   1   0∼ K_0.

We return to the original K_0 state.



§ CALCULATING THE CENTRAL CHARGE DIFFERENCE OF ONE-STEP CONDENSATION



In the one-step condensation from  to , the central charge is
changed by (M_c). In this section we give the detailed calculation.
Firstly, using the
modular relation for both  and , we find that 

    1/q_c√(|M_c|)   ∑_i,j,k∈∑_p=0^q_c|M_c|-1{S^_xiS^_ikT^_kkS^_kjS^_jy.
      
       .×exp[2π/2M_c(t_i+t_j-t_k+p)^2]}
       =exp( 2πc^-c^/8)T^_xxδ_xy.

To show 
c^-c^=(M_c)  8,
we need to use the reciprocity theorem for generalized
Gauss sums<cit.>:

    ∑_n=0^|c|-1^πan^2+bn/c=√(|c/a|)^π/4[(ac)-b^2/ac]∑_n=0^|a|-1^-πcn^2+bn/a,

where a,b,c are integers, ac≠ 0 and ac+b even. Thus,

    ∑_p=0^q_c|M_c|-1exp[2π/2M_c(t_i+t_j-t_k+p)^2]
      
       =1/q_c^π(t_i+t_j-t_k)^2/M_c∑_p=0^q_c^2|M_c|-1^π/M_cq_c^2[q_c^2p^2+2q_c^2(t_i+t_j-t_k)p]
       =√(|M_c|)/q_c^π/4(M_c)∑_p=0^q_c^2-1^-π[M_cp^2+2(t_i+t_j-t_k)p]
       =√(|M_c|)/q_c^π/4(M_c)∑_p=0^q_c^2-1^-π
      (m_c-2s_c)p^2S^_ia_c^⊗ p/S^_iS^_ja_c^⊗
    p/S^_jS^_ka_c^⊗ p/S^_k
       =√(|M_c|)/q_c^π/4(M_c)∑_p=0^q_c^2-1T^_a_c^⊗
      p,a_c^⊗ pS^_ia_c^⊗ p/S^_iS^_ja_c^⊗
    p/S^_jS^_ka_c^⊗ p/S^_k.

Substituting the above result into (<ref>), we have

    1/q_c√(|M_c|)∑_i,j,k∈∑_p=0^q_c|M_c|-1{S^_xiS^_ikT^_kkS^_kjS^_jy.
      
          .×exp[2π/2M_c(t_i+t_j-t_k+p)^2]}
       =1/q_c^2^π/4(M_c)∑_p=0^q_c^2-1∑_kT_kk^ T^_a_c^⊗ p,a_c^⊗ pS^_ka_c^⊗ p/S^_k
          ×∑_i S^_xiS^_ikS^_ia_c^⊗
      p/S^_i∑_j
      S^_kjS^_jyS^_ja_c^⊗ p/S^_j
       =1/q_c^2^π/4(M_c)∑_p=0^q_c^2-1∑_kT_k⊗ a_c^⊗ p,k⊗ a_c^⊗ p^ 
      N^k,a_c^⊗ p_x N^k,a_c^⊗ p_y
      
       =1/q_c^2^π/4(M_c)∑_p=0^q_c^2-1∑_kT_k⊗ a_c^⊗ p,k⊗ a_c^⊗ p^δ_k⊗ a_c^⊗ p,xδ_xy
       =^π/4(M_c)T^_xxδ_xy,

as desired.
In fact, based on the physical picture, we have a stronger result

    c^=c^+(M_c) .


So the central charge is changed by (M_c) after the one-step
condensation.  A direct corollary is that the central charge of an Abelian
topological order is given by the signature of its K-matrix (the number of
positive eigenvalues minus the number of negative eigenvalues).



§ THE GENERALIZED HIERARCHY CONSTRUCTION
AT FULL CATEGORICAL LEVEL



Does the generalized hierarchy construction from  to  described by
(<ref>), (<ref>), and (<ref>) always give a valid topological
order ?  To confirm this, below we will give a more rigorous construction
at full categorical level, which goes down to the level of F,R matrices.

The first step is to construct a pre-modular category , based on the
observation that the range of the second integer label can be reduced to
q_c|M_c|, and the combination m-t_i for (i,m) works as an gauge invariant
quantity. Such  can be viewed as a version of “semi-direct
product” of  with _q_c|M_c|. We use the gauge invariant m̃=m-t_i instead of the integer m to label anyons in ; in other
words, the anyons are labeled by the new pair (i,m̃) where i∈
and m̃+t_i ∈_q_c|M_c|. Fusion is then given by addition

    (i,m̃)⊗ (j,ñ)=⊕_k N^ij_k (k,[m̃+ñ]_q_c|M_c|),

where [⋯]_q_c|M_c| denotes the residue modulo q_c|M_c|. The F,R matrices in
 are given by those in  modified by appropriate phase
factors. More precisely, let F^i_1i_2i_3_i_4 

and R^i_1i_2_i_3 be the F,R matrices in ; then in  we take

    F̃^(i_1,m̃_1)(i_2,m̃_2)(i_3,m̃_3)_(i_4,m̃_4)
      =F^i_1i_2i_3_i_4^π/M_cm̃_1(m̃_2+m̃_3-[m̃_2+m̃_3]_q_c|M_c|),
       R̃^(i_1,m̃_1)(i_2,m̃_2)_(i_3,m̃_3)=R^i_1i_2_i_3^π/M_cm̃_1m̃_2
        =R^i_1i_2_i_3^π/M_c(m_1-t_i_1)(m_2-t_i_2).

It is straightforward to check that they satisfy the pentagon and hexagon
equations, and  is a valid pre-modular category. Moreover, the
modified R matrices do give us the desired modified spin. This also suggests
that the hierarchy construction equally works for pre-modular categories, thus
can be easily generalized to topological orders with
symmetries<cit.>.

The second step is to reduce the pre-modular category  to the
modular category .  Categorically, just note that {(i=a_c,m̃=M_c)^⊗ q,q=0,…,q_c-1} forms the Müger center of
, which can be identified with (_q_c); by condensing this
(_q_c) we obtain the desired modular category .  Put it simply,
we just further impose the equivalence relation (<ref>) in ,
such that one orbit of length q_c is viewed as one type of anyon instead of
q_c different types.  This way we rigorously recover the same construction
described by (<ref>), (<ref>), and (<ref>).




§ TABLES OF NON-ABELIAN FAMILIES



In this section, we list some non-Abelian families. 
Each table contains a family up to a certain N.
Each row corresponds to a topological order.  The anyons are listed with
increasing quantum dimensions.  Only the quantum dimensions of a root state is
explicitly given. The quantum dimensions of other topological orders can be
easily obtained from those of the root, since the degeneracy for each
value of quantum dimension scales linearly with N.
The anyons in red have trivial mutual statistics with all Abelian anyons.
Such sets of anyons are the same within each family, and is an invariant
of the non-Abelian family.

The following is the Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
1_0     1     0 1 

2_1     2     0,1/4  

2_7     2     0,3/4  

3_2     3     0,1/3,1/3  

3_6     3     0,2/3,2/3  

4_0     4     0,0,0,1/2  

4_0     4     0,0,1/4,3/4  

4_1     4     0,1/8,1/8,1/2  

4_7     4     0,7/8,7/8,1/2  

4_2     4     0,1/4,1/4,1/2  

4_6     4     0,3/4,3/4,1/2  

4_3     4     0,3/8,3/8,1/2  

4_5     4     0,5/8,5/8,1/2  

4_4     4     0,1/2,1/2,1/2  

5_0     5     0,1/5,1/5,4/5,4/5  

5_4     5     0,2/5,2/5,3/5,3/5  

6_1     6     0,1/12,1/12,3/4,1/3,1/3  

6_7     6     0,11/12,11/12,1/4,2/3,2/3  

6_3     6     0,1/4,1/3,1/3,7/12,7/12  

6_5     6     0,3/4,2/3,2/3,5/12,5/12  

7_2     7     0,1/7,1/7,2/7,2/7,4/7,4/7  

7_6     7     0,6/7,6/7,5/7,5/7,3/7,3/7  

8_0     8     0,1/8,1/8,7/8,7/8,1/4,3/4,1/2  

8_1     8     0,0,0,1/4,1/4,1/4,3/4,1/2  

8_1     8     0,0,1/16,1/16,1/4,1/4,9/16,9/16  

8_1     8     0,0,13/16,13/16,1/4,1/4,5/16,5/16  

8_7     8     0,0,0,1/4,3/4,3/4,3/4,1/2  

8_7     8     0,0,15/16,15/16,3/4,3/4,7/16,7/16  

8_7     8     0,0,3/16,3/16,3/4,3/4,11/16,11/16  

8_2     8     0,1/8,1/8,1/4,3/4,3/8,3/8,1/2  

8_6     8     0,7/8,7/8,1/4,3/4,5/8,5/8,1/2  

8_3     8     0,1/4,1/4,1/4,3/4,1/2,1/2,1/2  

8_5     8     0,1/4,3/4,3/4,3/4,1/2,1/2,1/2  

8_4     8     0,1/4,3/4,3/8,3/8,5/8,5/8,1/2  




 

The following non-Abelian family is described by effective SU(2)_-3
Chern-Simons (CS) theory plus some Abelian CS theories.  So it is called the
SU(2)_-3 non-Abelian family.  Due to the level-rank duality, it is also
called the SU(3)_2 non-Abelian family since its contains a state described
by SU(3)_2 CS theory. We can also call the family as the Fibonacci
non-Abelian family since the root state is the Fibonacci non-Abelian state.
This family contains FQH state <cit.>

    Ψ_ν=3/2({z_i})=[χ_3({z_i})]^2,   N_c =4_21/5



In general, for a state 

    Ψ_ν=k/n({ z_i})=[χ_k({ z_i})]^n,

its low energy effective theory obtained from the projective parton
construction is given by <cit.>

    = ψ_a^† (_0 _- (a_0)_ ) ψ_a
           
    - 
    |[(_i - A_i) _- (a_i)_ ]  ψ_a|^2/2m

where a=1,⋯,k=3 and =1,⋯,n=2, and a_μ is the SU(n) gauge
field doing the projection.  Before the projection (when a_μ=0) the
above effective theory describes a filling fraction ν=kn IQH state whose
edge has a chiral central charge c=kn (has kn right-moving modes).  After the  projection (after
integrating out the non-zero dynamical SU(n) gauge field a_μ), the edge
states will have a reduced central charge

    c = kn - k (n^2-1) /k+n.

For our case here, k=3 and n=2 and we find c=21/5 in
(<ref>).

If we integrate out that fermion fields ψ_a first, we will obtain an
effective SU(n) CS theory at level -k with central charge -
k (n^2-1) /k+n.  The state (<ref>) and the effective theory
(<ref>) has the same number of anyon types as the SU(n)_-k CS theory
But the spins of the anyons in (<ref>)
and in (<ref>) is not given by those of SU(n)_-k CS theory.  They
may differ by 1/2 since the anyons in (<ref>) may contain an extra
fermion field ψ_a.  So the spins of the anyons in (<ref>) and
in (<ref>) are related to the spins in SU(n)_-k CS theory via

    s_i = s_i^SU(n)_-k mod 1/2.

In other words, 
the spins of the anyons in (<ref>) and
in (<ref>) are related to the spins in SU(n)_k CS theory via

    s_i = - s_i^SU(n)_k mod 1/2.

This allows us to identify the state (<ref>) in the table of the
non-Abelian family which is marked by the red N_c.  We will denote the
effective SU(n) CS theory obtained from (<ref>) by integrating out the
fermions as SU(n)_-k^f.  So the red N_c=4_21/5 row in the
following table is described by the SU(n)_-k^f CS effective theory
(<ref>). On the other hand, the row N_c=4_31/5 is described by
the pure SU(n)_-k CS effective theory (without coupling to fermionic
fields).




1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
2_26/5     3.618     0,3/5 1,ζ_3^1 

4_31/5     7.236     0,1/4,17/20,3/5   SU(2)_-3 CS

4_21/5     7.236     0,3/4,7/20,3/5   SU(2)_-3^f CS

6_36/5     10.85     0,1/3,1/3,14/15,14/15,3/5  

6_16/5     10.85     0,2/3,2/3,4/15,4/15,3/5  

8_1/5     14.47     0,3/8,3/8,1/2,39/40,39/40,1/10,3/5  

8_36/5     14.47     0,1/4,1/4,1/2,1/10,17/20,17/20,3/5  

8_6/5     14.47     0,1/2,1/2,1/2,1/10,1/10,1/10,3/5  

8_31/5     14.47     0,1/8,1/8,1/2,1/10,29/40,29/40,3/5  

8_11/5     14.47     0,5/8,5/8,1/2,1/10,9/40,9/40,3/5  

8_26/5     14.47     0,0,1/4,3/4,17/20,7/20,3/5,3/5  

8_26/5     14.47     0,0,0,1/2,1/10,3/5,3/5,3/5  

8_16/5     14.47     0,3/4,3/4,1/2,1/10,7/20,7/20,3/5  

8_21/5     14.47     0,7/8,7/8,1/2,1/10,3/5,19/40,19/40  

10_6/5     18.09     0,2/5,2/5,3/5,3/5,0,0,1/5,1/5,3/5  

10_26/5     18.09     0,1/5,1/5,4/5,4/5,4/5,4/5,2/5,2/5,3/5  




 

The following non-Abelian family is described by effective SU(2)_3 CS theory.
So it is called the SU(2)_3 non-Abelian family.  The states in the following
table are time-reversal conjugates of those in the previous table.  This family
contains FQH state <cit.>

    Ψ_ν=-3/2({z̅_i})=[χ_3({z̅_i})]^2,  N_c =4_19/5



1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
2_14/5     3.618     0,2/5 1,ζ_3^1 

4_9/5     7.236     0,3/4,3/20,2/5  SU(2)_3 CS

4_19/5     7.236     0,1/4,13/20,2/5  SU(2)_3^f CS

6_4/5     10.85     0,2/3,2/3,1/15,1/15,2/5  

6_24/5     10.85     0,1/3,1/3,11/15,11/15,2/5  

8_39/5     14.47     0,5/8,5/8,1/2,1/40,1/40,9/10,2/5  

8_4/5     14.47     0,3/4,3/4,1/2,9/10,3/20,3/20,2/5  

8_34/5     14.47     0,1/2,1/2,1/2,9/10,9/10,9/10,2/5  

8_9/5     14.47     0,7/8,7/8,1/2,9/10,11/40,11/40,2/5  

8_29/5     14.47     0,3/8,3/8,1/2,9/10,31/40,31/40,2/5  

8_14/5     14.47     0,0,1/4,3/4,3/20,13/20,2/5,2/5  

8_14/5     14.47     0,0,0,1/2,9/10,2/5,2/5,2/5  

8_24/5     14.47     0,1/4,1/4,1/2,9/10,13/20,13/20,2/5  

8_19/5     14.47     0,1/8,1/8,1/2,9/10,2/5,21/40,21/40  

10_34/5     18.09     0,2/5,2/5,3/5,3/5,0,0,4/5,4/5,2/5  

10_14/5     18.09     0,1/5,1/5,4/5,4/5,1/5,1/5,2/5,3/5,3/5  






The following SU(2)_2 Ising non-Abelian family contains FQH states <cit.>

    Ψ_ν=1({z_i})=[χ_2({z_i})]^2,  N_c=3_5/2

and

    Ψ_ν=1 = (1/z_1-z_21/z_2-z_3⋯)∏ (z_i-z_j),
    ,  N_c=3_3/2
 

1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
3_1/2     4     0,1/2,1/16 1,1,ζ_2^1 

3_15/2     4     0,1/2,15/16  

3_3/2     4     0,1/2,3/16  

3_13/2     4     0,1/2,13/16  

3_5/2     4     0,1/2,5/16  

3_11/2     4     0,1/2,11/16  

3_7/2     4     0,1/2,7/16  

3_9/2     4     0,1/2,9/16  

6_1/2     8     0,1/4,3/4,1/2,15/16,3/16  

6_15/2     8     0,1/4,3/4,1/2,1/16,13/16  

6_3/2     8     0,1/4,3/4,1/2,1/16,5/16  

6_13/2     8     0,1/4,3/4,1/2,15/16,11/16  

6_5/2     8     0,1/4,3/4,1/2,3/16,7/16  

6_11/2     8     0,1/4,3/4,1/2,13/16,9/16  

6_7/2     8     0,1/4,3/4,1/2,5/16,9/16  

6_9/2     8     0,1/4,3/4,1/2,11/16,7/16  

9_1/2     12     0,5/6,5/6,1/3,1/3,1/2,7/48,7/48,13/16  

9_1/2     12     0,1/6,1/6,2/3,2/3,1/2,47/48,47/48,5/16  

9_15/2     12     0,5/6,5/6,1/3,1/3,1/2,1/48,1/48,11/16  

9_15/2     12     0,1/6,1/6,2/3,2/3,1/2,41/48,41/48,3/16  

9_3/2     12     0,5/6,5/6,1/3,1/3,1/2,15/16,13/48,13/48  

9_3/2     12     0,1/6,1/6,2/3,2/3,1/2,5/48,5/48,7/16  

9_13/2     12     0,5/6,5/6,1/3,1/3,1/2,43/48,43/48,9/16  

9_13/2     12     0,1/6,1/6,2/3,2/3,1/2,1/16,35/48,35/48  

9_5/2     12     0,5/6,5/6,1/3,1/3,1/2,1/16,19/48,19/48  

9_5/2     12     0,1/6,1/6,2/3,2/3,1/2,11/48,11/48,9/16  

9_11/2     12     0,5/6,5/6,1/3,1/3,1/2,37/48,37/48,7/16  

9_11/2     12     0,1/6,1/6,2/3,2/3,1/2,15/16,29/48,29/48  

9_7/2     12     0,5/6,5/6,1/3,1/3,1/2,3/16,25/48,25/48  

9_7/2     12     0,1/6,1/6,2/3,2/3,1/2,11/16,17/48,17/48  

9_9/2     12     0,5/6,5/6,1/3,1/3,1/2,5/16,31/48,31/48  

9_9/2     12     0,1/6,1/6,2/3,2/3,1/2,13/16,23/48,23/48  







The following SU(2)_5 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-5/2({z̅_i})=[χ_5({z̅_i})]^2,  N_c=6_1/7


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
3_8/7     9.295     0,6/7,2/7 1,ζ_5^1,ζ_5^2 

6_1/7     18.59     0,3/4,6/7,17/28,1/28,2/7  

6_15/7     18.59     0,1/4,3/28,6/7,2/7,15/28  

9_50/7     27.88     0,2/3,2/3,6/7,11/21,11/21,20/21,20/21,2/7  

9_22/7     27.88     0,1/3,1/3,6/7,4/21,4/21,2/7,13/21,13/21  

12_1/7     37.18     0,7/8,7/8,1/2,6/7,41/56,41/56,5/14,9/56,9/56,11/14,2/7  

12_50/7     37.18     0,3/4,3/4,1/2,6/7,5/14,17/28,17/28,1/28,1/28,11/14,2/7  

12_8/7     37.18     0,0,1/4,3/4,3/28,6/7,6/7,17/28,1/28,2/7,2/7,15/28  

12_8/7     37.18     0,0,0,1/2,6/7,6/7,6/7,5/14,11/14,2/7,2/7,2/7  

12_43/7     37.18     0,5/8,5/8,1/2,6/7,5/14,27/56,27/56,51/56,51/56,11/14,2/7  

12_15/7     37.18     0,1/8,1/8,1/2,55/56,55/56,6/7,5/14,11/14,2/7,23/56,23/56  

12_36/7     37.18     0,1/2,1/2,1/2,6/7,5/14,5/14,5/14,11/14,11/14,11/14,2/7  

12_22/7     37.18     0,1/4,1/4,1/2,3/28,3/28,6/7,5/14,11/14,2/7,15/28,15/28  

12_29/7     37.18     0,3/8,3/8,1/2,6/7,13/56,13/56,5/14,11/14,2/7,37/56,37/56  




 

 

The following SU(2)_-5 non-Abelian family (or SU(5)_2 non-Abelian family) contains FQH state <cit.>

    Ψ_ν=5/2({z_i})=[χ_5({z_i})]^2,  N_c=6_55/7


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
3_48/7     9.295     0,1/7,5/7 1,ζ_5^1,ζ_5^2 

6_55/7     18.59     0,1/4,1/7,11/28,27/28,5/7  

6_41/7     18.59     0,3/4,25/28,1/7,5/7,13/28  

9_6/7     27.88     0,1/3,1/3,1/7,10/21,10/21,1/21,1/21,5/7  

9_34/7     27.88     0,2/3,2/3,1/7,17/21,17/21,5/7,8/21,8/21  

12_55/7     37.18     0,1/8,1/8,1/2,1/7,15/56,15/56,9/14,47/56,47/56,3/14,5/7  

12_6/7     37.18     0,1/4,1/4,1/2,1/7,9/14,11/28,11/28,27/28,27/28,3/14,5/7  

12_48/7     37.18     0,0,1/4,3/4,25/28,1/7,1/7,11/28,27/28,5/7,5/7,13/28  

12_48/7     37.18     0,0,0,1/2,1/7,1/7,1/7,9/14,3/14,5/7,5/7,5/7  

12_13/7     37.18     0,3/8,3/8,1/2,1/7,9/14,29/56,29/56,5/56,5/56,3/14,5/7  

12_41/7     37.18     0,7/8,7/8,1/2,1/56,1/56,1/7,9/14,3/14,5/7,33/56,33/56  

12_20/7     37.18     0,1/2,1/2,1/2,1/7,9/14,9/14,9/14,3/14,3/14,3/14,5/7  

12_34/7     37.18     0,3/4,3/4,1/2,25/28,25/28,1/7,9/14,3/14,5/7,13/28,13/28  

12_27/7     37.18     0,5/8,5/8,1/2,1/7,43/56,43/56,9/14,3/14,5/7,19/56,19/56  






The following three  non-Abelian families are obtained
by stacking two FQH states from the two 
Fibonacci non-Abelian families.


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
4_0     13.09     0,2/5,3/5,0 1,ζ_3^1,ζ_3^1,ζ_8^2 

8_1     26.18     0,1/4,17/20,13/20,2/5,3/5,0,1/4  

8_7     26.18     0,3/4,3/20,7/20,2/5,3/5,0,3/4  

12_2     39.27     0,1/3,1/3,14/15,14/15,11/15,11/15,2/5,3/5,0,1/3,1/3  

12_6     39.27     0,2/3,2/3,1/15,1/15,4/15,4/15,2/5,3/5,0,2/3,2/3  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
4_12/5     13.09     0,3/5,3/5,1/5 1,ζ_3^1,ζ_3^1,ζ_8^2 

8_7/5     26.18     0,3/4,7/20,7/20,3/5,3/5,19/20,1/5  

8_17/5     26.18     0,1/4,17/20,17/20,3/5,3/5,1/5,9/20  

12_2/5     39.27     0,2/3,2/3,4/15,4/15,4/15,4/15,3/5,3/5,13/15,13/15,1/5  

12_22/5     39.27     0,1/3,1/3,14/15,14/15,14/15,14/15,3/5,3/5,1/5,8/15,8/15  





 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
4_28/5     13.09     0,2/5,2/5,4/5 1,ζ_3^1,ζ_3^1,ζ_8^2 

8_33/5     26.18     0,1/4,13/20,13/20,2/5,2/5,1/20,4/5  

8_23/5     26.18     0,3/4,3/20,3/20,2/5,2/5,4/5,11/20  

12_38/5     39.27     0,1/3,1/3,11/15,11/15,11/15,11/15,2/5,2/5,2/15,2/15,4/5  

12_18/5     39.27     0,2/3,2/3,1/15,1/15,1/15,1/15,2/5,2/5,4/5,7/15,7/15  




 

 

The following SU(2)_7 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-7/2({z̅_i})=[χ_7({z̅_i})]^2,  N_c=8_13/3


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
4_10/3     19.23     0,1/3,2/9,2/3 1,ζ_7^1,ζ_7^2,ζ_7^3 

8_7/3     38.46     0,3/4,1/12,1/3,35/36,2/9,2/3,5/12  

8_13/3     38.46     0,1/4,1/3,7/12,2/9,17/36,11/12,2/3  

12_4/3     57.70     0,2/3,2/3,0,0,1/3,8/9,8/9,2/9,1/3,1/3,2/3  

12_16/3     57.70     0,1/3,1/3,1/3,2/3,2/3,2/9,5/9,5/9,0,0,2/3  




 

 

The following SU(2)_-7 non-Abelian family contains FQH state <cit.>

    Ψ_ν=7/2({z_i})=[χ_7({z_i})]^2,  N_c=8_35/3


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
4_14/3     19.23     0,2/3,7/9,1/3 1,ζ_7^1,ζ_7^2,ζ_7^3 

8_17/3     38.46     0,1/4,11/12,2/3,1/36,7/9,1/3,7/12  

8_11/3     38.46     0,3/4,2/3,5/12,7/9,19/36,1/12,1/3  

12_20/3     57.70     0,1/3,1/3,0,0,2/3,1/9,1/9,7/9,1/3,2/3,2/3  

12_8/3     57.70     0,2/3,2/3,1/3,1/3,2/3,7/9,4/9,4/9,0,0,1/3  






The following SU(2)_4 non-Abelian family contains FQH state <cit.>

    Ψ_ν=1/2({z_i})=[χ_2({z_i})]^4,  N_c=10_3


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_2     12     0,0,1/8,5/8,1/3 1,1,ζ_4^1,ζ_4^1,2 

5_2     12     0,0,7/8,3/8,1/3  

10_1     24     0,0,3/4,3/4,1/8,7/8,3/8,5/8,1/12,1/3  

10_1     24     0,0,3/4,3/4,1/16,1/16,9/16,9/16,1/12,1/3  

10_1     24     0,0,3/4,3/4,13/16,13/16,5/16,5/16,1/12,1/3  

10_3     24     0,0,1/4,1/4,1/8,7/8,3/8,5/8,1/3,7/12  

10_3     24     0,0,1/4,1/4,3/16,3/16,11/16,11/16,1/3,7/12  SU(4)_-2^f, SU(4)_-2 

10_3     24     0,0,1/4,1/4,15/16,15/16,7/16,7/16,1/3,7/12  

15_0     36     0,0,2/3,2/3,2/3,2/3,1/24,1/24,7/8,3/8,13/24,13/24,0,0,1/3  

15_0     36     0,0,2/3,2/3,2/3,2/3,1/8,19/24,19/24,7/24,7/24,5/8,0,0,1/3  

15_4     36     0,0,1/3,1/3,1/3,1/3,23/24,23/24,1/8,5/8,11/24,11/24,1/3,2/3,2/3  

15_4     36     0,0,1/3,1/3,1/3,1/3,7/8,5/24,5/24,17/24,17/24,3/8,1/3,2/3,2/3  




 

 

The following SU(2)_-4 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-1/2({z̅_i})=[χ_2({z̅_i})]^4,  N_c=10_5


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_6     12     0,0,7/8,3/8,2/3 1,1,ζ_4^1,ζ_4^1,2 

5_6     12     0,0,1/8,5/8,2/3  

10_7     24     0,0,1/4,1/4,1/8,7/8,3/8,5/8,11/12,2/3  

10_7     24     0,0,1/4,1/4,15/16,15/16,7/16,7/16,11/12,2/3  

10_7     24     0,0,1/4,1/4,3/16,3/16,11/16,11/16,11/12,2/3  

10_5     24     0,0,3/4,3/4,1/8,7/8,3/8,5/8,2/3,5/12  

10_5     24     0,0,3/4,3/4,13/16,13/16,5/16,5/16,2/3,5/12  SU(4)_2^f, SU(4)_2 

10_5     24     0,0,3/4,3/4,1/16,1/16,9/16,9/16,2/3,5/12  

15_0     36     0,0,1/3,1/3,1/3,1/3,7/8,5/24,5/24,17/24,17/24,3/8,0,0,2/3  

15_0     36     0,0,1/3,1/3,1/3,1/3,23/24,23/24,1/8,5/8,11/24,11/24,0,0,2/3  

15_4     36     0,0,2/3,2/3,2/3,2/3,1/8,19/24,19/24,7/24,7/24,5/8,1/3,1/3,2/3  

15_4     36     0,0,2/3,2/3,2/3,2/3,1/24,1/24,7/8,3/8,13/24,13/24,1/3,1/3,2/3  




 

 

The following SU(2)_9 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-9/2({z̅_i})=[χ_9({z̅_i})]^2,  N_c=10_5/11


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_16/11     34.64     0,9/11,2/11,1/11,6/11 1,ζ_9^1,ζ_9^2,ζ_9^3,ζ_9^4 

10_5/11     69.29     0,3/4,9/11,25/44,41/44,2/11,1/11,37/44,13/44,6/11  

10_27/11     69.29     0,1/4,3/44,9/11,2/11,19/44,1/11,15/44,35/44,6/11  

15_82/11     103.9     0,2/3,2/3,9/11,16/33,16/33,28/33,28/33,2/11,1/11,25/33,25/33,7/33,7/33,6/11  

15_38/11     103.9     0,1/3,1/3,5/33,5/33,9/11,2/11,17/33,17/33,1/11,14/33,14/33,29/33,29/33,6/11  






The following SU(2)_-9 non-Abelian family contains FQH state <cit.>

    Ψ_ν=9/2({z_i})=[χ_9({z_i})]^2,  N_c=10_83/11


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_72/11     34.64     0,2/11,9/11,10/11,5/11 1,ζ_9^1,ζ_9^2,ζ_9^3,ζ_9^4 

10_83/11     69.29     0,1/4,2/11,19/44,3/44,9/11,10/11,7/44,31/44,5/11  

10_61/11     69.29     0,3/4,41/44,2/11,9/11,25/44,10/11,29/44,9/44,5/11  

15_6/11     103.9     0,1/3,1/3,2/11,17/33,17/33,5/33,5/33,9/11,10/11,8/33,8/33,26/33,26/33,5/11  

15_50/11     103.9     0,2/3,2/3,28/33,28/33,2/11,9/11,16/33,16/33,10/11,19/33,19/33,4/33,4/33,5/11  




 

 

The following SU(3)_4 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-7/4({z̅_i})=χ_1({z̅_i})[χ_4({z̅_i})]^3,  N_c=15_4/7


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_18/7     35.34     0,6/7,6/7,1/7,3/7 1,ζ_5^2,ζ_5^2,ζ_12^2,ζ_12^4 

10_11/7     70.68     0,3/4,6/7,6/7,17/28,17/28,25/28,1/7,5/28,3/7  

10_25/7     70.68     0,1/4,3/28,3/28,6/7,6/7,1/7,11/28,19/28,3/7  

15_4/7     106.0     0,2/3,2/3,6/7,6/7,11/21,11/21,11/21,11/21,1/7,17/21,17/21,2/21,2/21,3/7  

15_32/7     106.0     0,1/3,1/3,6/7,6/7,4/21,4/21,4/21,4/21,1/7,10/21,10/21,16/21,16/21,3/7  




 

 

The following SU(3)_-4 non-Abelian family contains FQH state <cit.>

    Ψ_ν=7/4({z_i})=χ_1({z_i})[χ_4({z_i})]^3,  N_c=15_52/7


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
5_38/7     35.34     0,1/7,1/7,6/7,4/7 1,ζ_5^2,ζ_5^2,ζ_12^2,ζ_12^4 

10_45/7     70.68     0,1/4,1/7,1/7,11/28,11/28,3/28,6/7,23/28,4/7  

10_31/7     70.68     0,3/4,25/28,25/28,1/7,1/7,6/7,17/28,9/28,4/7  

15_52/7     106.0     0,1/3,1/3,1/7,1/7,10/21,10/21,10/21,10/21,6/7,4/21,4/21,19/21,19/21,4/7  

15_24/7     106.0     0,2/3,2/3,1/7,1/7,17/21,17/21,17/21,17/21,6/7,11/21,11/21,5/21,5/21,4/7  








 

 

 

The following four non-Abelian families are obtained by stacking a FQH state
from the two SU(2)_± 3 non-Abelian families with a FQH state from the two
SU(2)_± 5 non-Abelian families.
 

1.25


N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_58/35     33.63     0,2/5,1/7,5/7,19/35,4/35 1,ζ_3^1,ζ_5^1,ζ_5^2,2.915,3.635 

12_23/35     67.26     0,3/4,3/20,2/5,25/28,1/7,5/7,13/28,41/140,19/35,4/35,121/140  

12_93/35     67.26     0,1/4,13/20,2/5,1/7,11/28,27/28,5/7,111/140,19/35,4/35,51/140  

18_268/35     100.8     0,2/3,2/3,1/15,1/15,2/5,1/7,17/21,17/21,5/7,8/21,8/21,22/105,22/105,19/35,4/35,82/105,82/105  

18_128/35     100.8     0,1/3,1/3,11/15,11/15,2/5,1/7,10/21,10/21,1/21,1/21,5/7,92/105,92/105,19/35,4/35,47/105,47/105  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_138/35     33.63     0,2/5,6/7,2/7,9/35,24/35 1,ζ_3^1,ζ_5^1,ζ_5^2,2.915,3.635 

12_103/35     67.26     0,3/4,3/20,2/5,6/7,17/28,1/28,2/7,1/140,9/35,24/35,61/140  

12_173/35     67.26     0,1/4,13/20,2/5,3/28,6/7,2/7,15/28,9/35,71/140,131/140,24/35  

18_68/35     100.8     0,2/3,2/3,1/15,1/15,2/5,6/7,11/21,11/21,20/21,20/21,2/7,97/105,97/105,9/35,24/35,37/105,37/105  

18_208/35     100.8     0,1/3,1/3,11/15,11/15,2/5,6/7,4/21,4/21,2/7,13/21,13/21,9/35,62/105,62/105,2/105,2/105,24/35  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_142/35     33.63     0,3/5,1/7,5/7,26/35,11/35 1,ζ_3^1,ζ_5^1,ζ_5^2,2.915,3.635 

12_177/35     67.26     0,1/4,17/20,3/5,1/7,11/28,27/28,5/7,139/140,26/35,11/35,79/140  

12_107/35     67.26     0,3/4,7/20,3/5,25/28,1/7,5/7,13/28,26/35,69/140,9/140,11/35  

18_212/35     100.8     0,1/3,1/3,14/15,14/15,3/5,1/7,10/21,10/21,1/21,1/21,5/7,8/105,8/105,26/35,11/35,68/105,68/105  

18_72/35     100.8     0,2/3,2/3,4/15,4/15,3/5,1/7,17/21,17/21,5/7,8/21,8/21,26/35,43/105,43/105,103/105,103/105,11/35  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_222/35     33.63     0,3/5,6/7,2/7,16/35,31/35 1,ζ_3^1,ζ_5^1,ζ_5^2,2.915,3.635 

12_257/35     67.26     0,1/4,17/20,3/5,3/28,6/7,2/7,15/28,99/140,16/35,31/35,19/140  

12_187/35     67.26     0,3/4,7/20,3/5,6/7,17/28,1/28,2/7,29/140,16/35,31/35,89/140  

18_12/35     100.8     0,1/3,1/3,14/15,14/15,3/5,6/7,4/21,4/21,2/7,13/21,13/21,83/105,83/105,16/35,31/35,23/105,23/105  

18_152/35     100.8     0,2/3,2/3,4/15,4/15,3/5,6/7,11/21,11/21,20/21,20/21,2/7,13/105,13/105,16/35,31/35,58/105,58/105  






The following two non-Abelian families are obtained by stacking a FQH state
from the two Fibonacci non-Abelian families with a FQH state from the Ising
non-Abelian family.


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_3/10     14.47     0,1/2,11/16,9/10,2/5,7/80 1,1,ζ_2^1,ζ_3^1,ζ_3^1,2.288 

6_73/10     14.47     0,1/2,9/16,9/10,2/5,77/80  

6_13/10     14.47     0,1/2,13/16,9/10,2/5,17/80  

6_63/10     14.47     0,1/2,7/16,9/10,2/5,67/80  

6_23/10     14.47     0,1/2,15/16,9/10,2/5,27/80  

6_53/10     14.47     0,1/2,5/16,9/10,2/5,57/80  

6_33/10     14.47     0,1/2,1/16,9/10,2/5,37/80  

6_43/10     14.47     0,1/2,3/16,9/10,2/5,47/80  

12_3/10     28.94     0,1/4,3/4,1/2,13/16,9/16,9/10,3/20,13/20,2/5,77/80,17/80  

12_73/10     28.94     0,1/4,3/4,1/2,11/16,7/16,9/10,3/20,13/20,2/5,7/80,67/80  

12_13/10     28.94     0,1/4,3/4,1/2,15/16,11/16,9/10,3/20,13/20,2/5,7/80,27/80  

12_63/10     28.94     0,1/4,3/4,1/2,5/16,9/16,9/10,3/20,13/20,2/5,77/80,57/80  

12_23/10     28.94     0,1/4,3/4,1/2,1/16,13/16,9/10,3/20,13/20,2/5,17/80,37/80  

12_53/10     28.94     0,1/4,3/4,1/2,3/16,7/16,9/10,3/20,13/20,2/5,67/80,47/80  

12_33/10     28.94     0,1/4,3/4,1/2,15/16,3/16,9/10,3/20,13/20,2/5,27/80,47/80  

12_43/10     28.94     0,1/4,3/4,1/2,1/16,5/16,9/10,3/20,13/20,2/5,57/80,37/80  

18_3/10     43.41     0,1/6,1/6,2/3,2/3,1/2,15/16,29/48,29/48,1/15,1/15,9/10,2/5,17/30,17/30,1/240,1/240,27/80  

18_3/10     43.41     0,5/6,5/6,1/3,1/3,1/2,37/48,37/48,7/16,9/10,7/30,7/30,11/15,11/15,2/5,67/80,41/240,41/240  

18_73/10     43.41     0,5/6,5/6,1/3,1/3,1/2,5/16,31/48,31/48,9/10,7/30,7/30,11/15,11/15,2/5,11/240,11/240,57/80  

18_73/10     43.41     0,1/6,1/6,2/3,2/3,1/2,13/16,23/48,23/48,1/15,1/15,9/10,2/5,17/30,17/30,211/240,211/240,17/80  

18_13/10     43.41     0,1/6,1/6,2/3,2/3,1/2,1/16,35/48,35/48,1/15,1/15,9/10,2/5,17/30,17/30,31/240,31/240,37/80  

18_13/10     43.41     0,5/6,5/6,1/3,1/3,1/2,43/48,43/48,9/16,9/10,7/30,7/30,11/15,11/15,2/5,77/80,71/240,71/240  

18_63/10     43.41     0,5/6,5/6,1/3,1/3,1/2,3/16,25/48,25/48,9/10,7/30,7/30,11/15,11/15,2/5,221/240,221/240,47/80  

18_63/10     43.41     0,1/6,1/6,2/3,2/3,1/2,11/16,17/48,17/48,1/15,1/15,9/10,2/5,17/30,17/30,7/80,181/240,181/240  

18_23/10     43.41     0,5/6,5/6,1/3,1/3,1/2,1/48,1/48,11/16,9/10,7/30,7/30,11/15,11/15,2/5,7/80,101/240,101/240  

18_23/10     43.41     0,1/6,1/6,2/3,2/3,1/2,41/48,41/48,3/16,1/15,1/15,9/10,2/5,17/30,17/30,61/240,61/240,47/80  

18_53/10     43.41     0,5/6,5/6,1/3,1/3,1/2,1/16,19/48,19/48,9/10,7/30,7/30,11/15,11/15,2/5,191/240,191/240,37/80  

18_53/10     43.41     0,1/6,1/6,2/3,2/3,1/2,11/48,11/48,9/16,1/15,1/15,9/10,2/5,17/30,17/30,77/80,151/240,151/240  

18_33/10     43.41     0,5/6,5/6,1/3,1/3,1/2,7/48,7/48,13/16,9/10,7/30,7/30,11/15,11/15,2/5,17/80,131/240,131/240  

18_33/10     43.41     0,1/6,1/6,2/3,2/3,1/2,47/48,47/48,5/16,1/15,1/15,9/10,2/5,17/30,17/30,57/80,91/240,91/240  

18_43/10     43.41     0,5/6,5/6,1/3,1/3,1/2,15/16,13/48,13/48,9/10,7/30,7/30,11/15,11/15,2/5,161/240,161/240,27/80  

18_43/10     43.41     0,1/6,1/6,2/3,2/3,1/2,5/48,5/48,7/16,1/15,1/15,9/10,2/5,17/30,17/30,67/80,121/240,121/240  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_77/10     14.47     0,1/2,5/16,1/10,3/5,73/80 1,1,ζ_2^1,ζ_3^1,ζ_3^1,2.288 

6_7/10     14.47     0,1/2,7/16,1/10,3/5,3/80  

6_67/10     14.47     0,1/2,3/16,1/10,3/5,63/80  

6_17/10     14.47     0,1/2,9/16,1/10,3/5,13/80  

6_57/10     14.47     0,1/2,1/16,1/10,3/5,53/80  

6_27/10     14.47     0,1/2,11/16,1/10,3/5,23/80  

6_47/10     14.47     0,1/2,15/16,1/10,3/5,43/80  

6_37/10     14.47     0,1/2,13/16,1/10,3/5,33/80  

12_77/10     28.94     0,1/4,3/4,1/2,3/16,7/16,1/10,17/20,7/20,3/5,3/80,63/80  

12_7/10     28.94     0,1/4,3/4,1/2,5/16,9/16,1/10,17/20,7/20,3/5,73/80,13/80  

12_67/10     28.94     0,1/4,3/4,1/2,1/16,5/16,1/10,17/20,7/20,3/5,73/80,53/80  

12_17/10     28.94     0,1/4,3/4,1/2,11/16,7/16,1/10,17/20,7/20,3/5,3/80,23/80  

12_57/10     28.94     0,1/4,3/4,1/2,15/16,3/16,1/10,17/20,7/20,3/5,63/80,43/80  

12_27/10     28.94     0,1/4,3/4,1/2,13/16,9/16,1/10,17/20,7/20,3/5,13/80,33/80  

12_47/10     28.94     0,1/4,3/4,1/2,1/16,13/16,1/10,17/20,7/20,3/5,53/80,33/80  

12_37/10     28.94     0,1/4,3/4,1/2,15/16,11/16,1/10,17/20,7/20,3/5,23/80,43/80  

18_77/10     43.41     0,1/6,1/6,2/3,2/3,1/2,11/48,11/48,9/16,1/10,23/30,23/30,4/15,4/15,3/5,13/80,199/240,199/240  

18_77/10     43.41     0,5/6,5/6,1/3,1/3,1/2,1/16,19/48,19/48,14/15,14/15,1/10,3/5,13/30,13/30,239/240,239/240,53/80  

18_7/10     43.41     0,5/6,5/6,1/3,1/3,1/2,3/16,25/48,25/48,14/15,14/15,1/10,3/5,13/30,13/30,29/240,29/240,63/80  

18_7/10     43.41     0,1/6,1/6,2/3,2/3,1/2,11/16,17/48,17/48,1/10,23/30,23/30,4/15,4/15,3/5,229/240,229/240,23/80  

18_67/10     43.41     0,1/6,1/6,2/3,2/3,1/2,5/48,5/48,7/16,1/10,23/30,23/30,4/15,4/15,3/5,3/80,169/240,169/240  

18_67/10     43.41     0,5/6,5/6,1/3,1/3,1/2,15/16,13/48,13/48,14/15,14/15,1/10,3/5,13/30,13/30,209/240,209/240,43/80  

18_17/10     43.41     0,5/6,5/6,1/3,1/3,1/2,5/16,31/48,31/48,14/15,14/15,1/10,3/5,13/30,13/30,73/80,59/240,59/240  

18_17/10     43.41     0,1/6,1/6,2/3,2/3,1/2,13/16,23/48,23/48,1/10,23/30,23/30,4/15,4/15,3/5,19/240,19/240,33/80  

18_57/10     43.41     0,1/6,1/6,2/3,2/3,1/2,47/48,47/48,5/16,1/10,23/30,23/30,4/15,4/15,3/5,73/80,139/240,139/240  

18_57/10     43.41     0,5/6,5/6,1/3,1/3,1/2,7/48,7/48,13/16,14/15,14/15,1/10,3/5,13/30,13/30,179/240,179/240,33/80  

18_27/10     43.41     0,5/6,5/6,1/3,1/3,1/2,37/48,37/48,7/16,14/15,14/15,1/10,3/5,13/30,13/30,3/80,89/240,89/240  

18_27/10     43.41     0,1/6,1/6,2/3,2/3,1/2,15/16,29/48,29/48,1/10,23/30,23/30,4/15,4/15,3/5,49/240,49/240,43/80  

18_47/10     43.41     0,1/6,1/6,2/3,2/3,1/2,41/48,41/48,3/16,1/10,23/30,23/30,4/15,4/15,3/5,63/80,109/240,109/240  

18_47/10     43.41     0,5/6,5/6,1/3,1/3,1/2,1/48,1/48,11/16,14/15,14/15,1/10,3/5,13/30,13/30,23/80,149/240,149/240  

18_37/10     43.41     0,5/6,5/6,1/3,1/3,1/2,43/48,43/48,9/16,14/15,14/15,1/10,3/5,13/30,13/30,13/80,119/240,119/240  

18_37/10     43.41     0,1/6,1/6,2/3,2/3,1/2,1/16,35/48,35/48,1/10,23/30,23/30,4/15,4/15,3/5,79/240,79/240,53/80  






𝔰𝔬(10)_2 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_0     20     0,0,1/5,4/5,0,1/2 1,1,2,2,√(5),√(5) 

6_0     20     0,0,1/5,4/5,1/4,3/4  

12_1     40     0,0,1/4,1/4,1/20,1/5,4/5,9/20,0,1/4,3/4,1/2  

12_1     40     0,0,1/4,1/4,1/20,1/5,4/5,9/20,1/16,1/16,9/16,9/16  

12_1     40     0,0,1/4,1/4,1/20,1/5,4/5,9/20,13/16,13/16,5/16,5/16  

12_7     40     0,0,3/4,3/4,19/20,1/5,4/5,11/20,0,1/4,3/4,1/2  

12_7     40     0,0,3/4,3/4,19/20,1/5,4/5,11/20,15/16,15/16,7/16,7/16  

12_7     40     0,0,3/4,3/4,19/20,1/5,4/5,11/20,3/16,3/16,11/16,11/16  

18_2     60     0,0,1/3,1/3,1/3,1/3,2/15,2/15,1/5,4/5,8/15,8/15,0,5/6,5/6,1/3,1/3,1/2  

18_2     60     0,0,1/3,1/3,1/3,1/3,2/15,2/15,1/5,4/5,8/15,8/15,1/12,1/12,1/4,3/4,7/12,7/12  

18_6     60     0,0,2/3,2/3,2/3,2/3,13/15,13/15,1/5,4/5,7/15,7/15,11/12,11/12,1/4,3/4,5/12,5/12  

18_6     60     0,0,2/3,2/3,2/3,2/3,13/15,13/15,1/5,4/5,7/15,7/15,0,1/6,1/6,2/3,2/3,1/2  




 

𝔰𝔬(5)_2 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_4     20     0,0,2/5,3/5,1/4,3/4 1,1,2,2,√(5),√(5) 

6_4     20     0,0,2/5,3/5,0,1/2  

12_3     40     0,0,3/4,3/4,3/20,7/20,2/5,3/5,0,1/4,3/4,1/2  

12_3     40     0,0,3/4,3/4,3/20,7/20,2/5,3/5,3/16,3/16,11/16,11/16  

12_3     40     0,0,3/4,3/4,3/20,7/20,2/5,3/5,15/16,15/16,7/16,7/16  

12_5     40     0,0,1/4,1/4,17/20,13/20,2/5,3/5,0,1/4,3/4,1/2  

12_5     40     0,0,1/4,1/4,17/20,13/20,2/5,3/5,13/16,13/16,5/16,5/16  

12_5     40     0,0,1/4,1/4,17/20,13/20,2/5,3/5,1/16,1/16,9/16,9/16  

18_2     60     0,0,2/3,2/3,2/3,2/3,1/15,1/15,4/15,4/15,2/5,3/5,0,1/6,1/6,2/3,2/3,1/2  

18_2     60     0,0,2/3,2/3,2/3,2/3,1/15,1/15,4/15,4/15,2/5,3/5,11/12,11/12,1/4,3/4,5/12,5/12  

18_6     60     0,0,1/3,1/3,1/3,1/3,14/15,14/15,11/15,11/15,2/5,3/5,1/12,1/12,1/4,3/4,7/12,7/12  

18_6     60     0,0,1/3,1/3,1/3,1/3,14/15,14/15,11/15,11/15,2/5,3/5,0,5/6,5/6,1/3,1/3,1/2  





 

The following SU(2)_11 non-Abelian family contains FQH state <cit.>

    Ψ_ν=-11/2({z̅_i})=[χ_11({z̅_i})]^2,  N_c=12_59/13


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_46/13     56.74     0,4/13,2/13,7/13,6/13,12/13 1,ζ_11^1,ζ_11^2,ζ_11^3,ζ_11^4,ζ_11^5 

12_33/13     113.4     0,3/4,3/52,4/13,47/52,2/13,15/52,7/13,11/52,6/13,12/13,35/52  

12_59/13     113.4     0,1/4,4/13,29/52,2/13,21/52,41/52,7/13,37/52,6/13,12/13,9/52  

18_20/13     170.2     0,2/3,2/3,38/39,38/39,4/13,2/13,32/39,32/39,8/39,8/39,7/13,5/39,5/39,6/13,12/13,23/39,23/39  

18_72/13     170.2     0,1/3,1/3,4/13,25/39,25/39,2/13,19/39,19/39,34/39,34/39,7/13,31/39,31/39,6/13,12/13,10/39,10/39  




 


The following SU(2)_-11 non-Abelian family contains FQH state <cit.>

    Ψ_ν=11/2({z_i})=[χ_11({z_i})]^2,  N_c=12_45/13


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_58/13     56.74     0,9/13,11/13,6/13,7/13,1/13 1,ζ_11^1,ζ_11^2,ζ_11^3,ζ_11^4,ζ_11^5 

12_71/13     113.4     0,1/4,49/52,9/13,5/52,11/13,37/52,6/13,41/52,7/13,1/13,17/52  

12_45/13     113.4     0,3/4,9/13,23/52,11/13,31/52,11/52,6/13,15/52,7/13,1/13,43/52  

18_84/13     170.2     0,1/3,1/3,1/39,1/39,9/13,11/13,7/39,7/39,31/39,31/39,6/13,34/39,34/39,7/13,1/13,16/39,16/39  

18_32/13     170.2     0,2/3,2/3,9/13,14/39,14/39,11/13,20/39,20/39,5/39,5/39,6/13,8/39,8/39,7/13,1/13,29/39,29/39  




 

The following is the 𝔰𝔬(8)_-3 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_8/3     74.61     0,1/9,1/9,1/9,1/3,2/3 1,ζ_7^3,ζ_7^3,ζ_7^3,ζ_16^4,ζ_16^6 

12_5/3     149.2     0,3/4,1/9,1/9,1/9,31/36,31/36,31/36,1/12,1/3,2/3,5/12  

12_11/3     149.2     0,1/4,1/9,1/9,1/9,13/36,13/36,13/36,1/3,7/12,11/12,2/3  

18_2/3     223.8     0,2/3,2/3,1/9,1/9,1/9,7/9,7/9,7/9,7/9,7/9,7/9,0,0,1/3,1/3,1/3,2/3  

18_14/3     223.8     0,1/3,1/3,1/9,1/9,1/9,4/9,4/9,4/9,4/9,4/9,4/9,1/3,2/3,2/3,0,0,2/3  




 

The following is the 𝔰𝔬(8)_3 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_16/3     74.61     0,8/9,8/9,8/9,2/3,1/3 1,ζ_7^3,ζ_7^3,ζ_7^3,ζ_16^4,ζ_16^6 

12_19/3     149.2     0,1/4,8/9,8/9,8/9,5/36,5/36,5/36,11/12,2/3,1/3,7/12  

12_13/3     149.2     0,3/4,8/9,8/9,8/9,23/36,23/36,23/36,2/3,5/12,1/12,1/3  

18_22/3     223.8     0,1/3,1/3,8/9,8/9,8/9,2/9,2/9,2/9,2/9,2/9,2/9,0,0,2/3,1/3,2/3,2/3  

18_10/3     223.8     0,2/3,2/3,8/9,8/9,8/9,5/9,5/9,5/9,5/9,5/9,5/9,1/3,1/3,2/3,0,0,1/3  




 

The following is the (E_7)_3 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_2     100.6     0,6/7,5/7,3/7,0,1/3 1,3+√(21)/2,3+√(21)/2,3+√(21)/2,5+√(21)/2,7+√(21)/2 

12_1     201.2     0,3/4,6/7,5/28,5/7,17/28,3/7,13/28,0,3/4,1/12,1/3  

12_3     201.2     0,1/4,27/28,3/28,6/7,5/7,19/28,3/7,0,1/4,1/3,7/12  

18_0     301.8     0,2/3,2/3,2/21,2/21,6/7,5/7,8/21,8/21,3/7,11/21,11/21,0,2/3,2/3,0,0,1/3  

18_4     301.8     0,1/3,1/3,1/21,1/21,6/7,4/21,4/21,16/21,16/21,5/7,3/7,0,1/3,1/3,1/3,2/3,2/3  




 

The following is the (E_7)_-3 non-Abelian family:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
6_6     100.6     0,1/7,2/7,4/7,0,2/3 1,3+√(21)/2,3+√(21)/2,3+√(21)/2,5+√(21)/2,7+√(21)/2 

12_7     201.2     0,1/4,1/7,23/28,2/7,11/28,4/7,15/28,0,1/4,11/12,2/3  

12_5     201.2     0,3/4,1/28,25/28,1/7,2/7,9/28,4/7,0,3/4,2/3,5/12  

18_0     301.8     0,1/3,1/3,19/21,19/21,1/7,2/7,13/21,13/21,4/7,10/21,10/21,0,1/3,1/3,0,0,2/3  

18_4     301.8     0,2/3,2/3,20/21,20/21,1/7,17/21,17/21,5/21,5/21,2/7,4/7,0,2/3,2/3,1/3,1/3,2/3  




The following are other non-Abelian families:


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_1/4     27.31     0,1/2,27/32,27/32,1/4,3/4,7/32 1,1,ζ_6^1,ζ_6^1,ζ_6^2,ζ_6^2,ζ_6^3 

7_29/4     27.31     0,1/2,23/32,23/32,1/4,3/4,3/32  

7_5/4     27.31     0,1/2,31/32,31/32,1/4,3/4,11/32  

7_25/4     27.31     0,1/2,19/32,19/32,1/4,3/4,31/32  

7_9/4     27.31     0,1/2,3/32,3/32,1/4,3/4,15/32  

7_21/4     27.31     0,1/2,15/32,15/32,1/4,3/4,27/32  

7_13/4     27.31     0,1/2,7/32,7/32,1/4,3/4,19/32  

7_17/4     27.31     0,1/2,11/32,11/32,1/4,3/4,23/32  

14_1/4     54.62     0,1/4,3/4,1/2,31/32,31/32,23/32,23/32,0,1/4,3/4,1/2,3/32,11/32  

14_29/4     54.62     0,1/4,3/4,1/2,27/32,27/32,19/32,19/32,0,1/4,3/4,1/2,31/32,7/32  

14_5/4     54.62     0,1/4,3/4,1/2,3/32,3/32,27/32,27/32,0,1/4,3/4,1/2,7/32,15/32  

14_25/4     54.62     0,1/4,3/4,1/2,23/32,23/32,15/32,15/32,0,1/4,3/4,1/2,3/32,27/32  

14_9/4     54.62     0,1/4,3/4,1/2,31/32,31/32,7/32,7/32,0,1/4,3/4,1/2,11/32,19/32  

14_21/4     54.62     0,1/4,3/4,1/2,11/32,11/32,19/32,19/32,0,1/4,3/4,1/2,31/32,23/32  

14_13/4     54.62     0,1/4,3/4,1/2,3/32,3/32,11/32,11/32,0,1/4,3/4,1/2,23/32,15/32  

14_17/4     54.62     0,1/4,3/4,1/2,7/32,7/32,15/32,15/32,0,1/4,3/4,1/2,27/32,19/32  

21_1/4     81.94     0,5/6,5/6,1/3,1/3,1/2,89/96,89/96,89/96,89/96,19/32,19/32,1/12,1/12,1/4,3/4,7/12,7/12,31/32,29/96,29/96  

21_1/4     81.94     0,1/6,1/6,2/3,2/3,1/2,3/32,3/32,73/96,73/96,73/96,73/96,11/12,11/12,1/4,3/4,5/12,5/12,13/96,13/96,15/32  

21_29/4     81.94     0,1/6,1/6,2/3,2/3,1/2,31/32,31/32,61/96,61/96,61/96,61/96,11/12,11/12,1/4,3/4,5/12,5/12,1/96,1/96,11/32  

21_29/4     81.94     0,5/6,5/6,1/3,1/3,1/2,77/96,77/96,77/96,77/96,15/32,15/32,1/12,1/12,1/4,3/4,7/12,7/12,27/32,17/96,17/96  

21_5/4     81.94     0,5/6,5/6,1/3,1/3,1/2,5/96,5/96,5/96,5/96,23/32,23/32,1/12,1/12,1/4,3/4,7/12,7/12,3/32,41/96,41/96  

21_5/4     81.94     0,1/6,1/6,2/3,2/3,1/2,85/96,85/96,85/96,85/96,7/32,7/32,11/12,11/12,1/4,3/4,5/12,5/12,25/96,25/96,19/32  

21_25/4     81.94     0,1/6,1/6,2/3,2/3,1/2,27/32,27/32,49/96,49/96,49/96,49/96,11/12,11/12,1/4,3/4,5/12,5/12,85/96,85/96,7/32  

21_25/4     81.94     0,5/6,5/6,1/3,1/3,1/2,65/96,65/96,65/96,65/96,11/32,11/32,1/12,1/12,1/4,3/4,7/12,7/12,5/96,5/96,23/32  

21_9/4     81.94     0,1/6,1/6,2/3,2/3,1/2,1/96,1/96,1/96,1/96,11/32,11/32,11/12,11/12,1/4,3/4,5/12,5/12,23/32,37/96,37/96  

21_9/4     81.94     0,5/6,5/6,1/3,1/3,1/2,27/32,27/32,17/96,17/96,17/96,17/96,1/12,1/12,1/4,3/4,7/12,7/12,7/32,53/96,53/96  

21_21/4     81.94     0,5/6,5/6,1/3,1/3,1/2,7/32,7/32,53/96,53/96,53/96,53/96,1/12,1/12,1/4,3/4,7/12,7/12,89/96,89/96,19/32  

21_21/4     81.94     0,1/6,1/6,2/3,2/3,1/2,23/32,23/32,37/96,37/96,37/96,37/96,11/12,11/12,1/4,3/4,5/12,5/12,3/32,73/96,73/96  

21_13/4     81.94     0,1/6,1/6,2/3,2/3,1/2,13/96,13/96,13/96,13/96,15/32,15/32,11/12,11/12,1/4,3/4,5/12,5/12,27/32,49/96,49/96  

21_13/4     81.94     0,5/6,5/6,1/3,1/3,1/2,31/32,31/32,29/96,29/96,29/96,29/96,1/12,1/12,1/4,3/4,7/12,7/12,65/96,65/96,11/32  

21_17/4     81.94     0,5/6,5/6,1/3,1/3,1/2,3/32,3/32,41/96,41/96,41/96,41/96,1/12,1/12,1/4,3/4,7/12,7/12,77/96,77/96,15/32  

21_17/4     81.94     0,1/6,1/6,2/3,2/3,1/2,25/96,25/96,25/96,25/96,19/32,19/32,11/12,11/12,1/4,3/4,5/12,5/12,31/32,61/96,61/96  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_31/4     27.31     0,1/2,5/32,5/32,1/4,3/4,25/32 1,1,ζ_6^1,ζ_6^1,ζ_6^2,ζ_6^2,ζ_6^3 

7_3/4     27.31     0,1/2,9/32,9/32,1/4,3/4,29/32  

7_27/4     27.31     0,1/2,1/32,1/32,1/4,3/4,21/32  

7_7/4     27.31     0,1/2,13/32,13/32,1/4,3/4,1/32  

7_23/4     27.31     0,1/2,29/32,29/32,1/4,3/4,17/32  

7_11/4     27.31     0,1/2,17/32,17/32,1/4,3/4,5/32  

7_19/4     27.31     0,1/2,25/32,25/32,1/4,3/4,13/32  

7_15/4     27.31     0,1/2,21/32,21/32,1/4,3/4,9/32  

14_31/4     54.62     0,1/4,3/4,1/2,1/32,1/32,9/32,9/32,0,1/4,3/4,1/2,29/32,21/32  

14_3/4     54.62     0,1/4,3/4,1/2,5/32,5/32,13/32,13/32,0,1/4,3/4,1/2,1/32,25/32  

14_27/4     54.62     0,1/4,3/4,1/2,29/32,29/32,5/32,5/32,0,1/4,3/4,1/2,25/32,17/32  

14_7/4     54.62     0,1/4,3/4,1/2,9/32,9/32,17/32,17/32,0,1/4,3/4,1/2,29/32,5/32  

14_23/4     54.62     0,1/4,3/4,1/2,1/32,1/32,25/32,25/32,0,1/4,3/4,1/2,21/32,13/32  

14_11/4     54.62     0,1/4,3/4,1/2,21/32,21/32,13/32,13/32,0,1/4,3/4,1/2,1/32,9/32  

14_19/4     54.62     0,1/4,3/4,1/2,29/32,29/32,21/32,21/32,0,1/4,3/4,1/2,9/32,17/32  

14_15/4     54.62     0,1/4,3/4,1/2,25/32,25/32,17/32,17/32,0,1/4,3/4,1/2,5/32,13/32  

21_31/4     81.94     0,5/6,5/6,1/3,1/3,1/2,29/32,29/32,23/96,23/96,23/96,23/96,1/12,1/12,1/4,3/4,7/12,7/12,83/96,83/96,17/32  

21_31/4     81.94     0,1/6,1/6,2/3,2/3,1/2,7/96,7/96,7/96,7/96,13/32,13/32,11/12,11/12,1/4,3/4,5/12,5/12,1/32,67/96,67/96  

21_3/4     81.94     0,5/6,5/6,1/3,1/3,1/2,1/32,1/32,35/96,35/96,35/96,35/96,1/12,1/12,1/4,3/4,7/12,7/12,95/96,95/96,21/32  

21_3/4     81.94     0,1/6,1/6,2/3,2/3,1/2,19/96,19/96,19/96,19/96,17/32,17/32,11/12,11/12,1/4,3/4,5/12,5/12,5/32,79/96,79/96  

21_27/4     81.94     0,1/6,1/6,2/3,2/3,1/2,91/96,91/96,91/96,91/96,9/32,9/32,11/12,11/12,1/4,3/4,5/12,5/12,29/32,55/96,55/96  

21_27/4     81.94     0,5/6,5/6,1/3,1/3,1/2,11/96,11/96,11/96,11/96,25/32,25/32,1/12,1/12,1/4,3/4,7/12,7/12,71/96,71/96,13/32  

21_7/4     81.94     0,1/6,1/6,2/3,2/3,1/2,31/96,31/96,31/96,31/96,21/32,21/32,11/12,11/12,1/4,3/4,5/12,5/12,91/96,91/96,9/32  

21_7/4     81.94     0,5/6,5/6,1/3,1/3,1/2,5/32,5/32,47/96,47/96,47/96,47/96,1/12,1/12,1/4,3/4,7/12,7/12,11/96,11/96,25/32  

21_23/4     81.94     0,1/6,1/6,2/3,2/3,1/2,5/32,5/32,79/96,79/96,79/96,79/96,11/12,11/12,1/4,3/4,5/12,5/12,25/32,43/96,43/96  

21_23/4     81.94     0,5/6,5/6,1/3,1/3,1/2,95/96,95/96,95/96,95/96,21/32,21/32,1/12,1/12,1/4,3/4,7/12,7/12,9/32,59/96,59/96  

21_11/4     81.94     0,1/6,1/6,2/3,2/3,1/2,25/32,25/32,43/96,43/96,43/96,43/96,11/12,11/12,1/4,3/4,5/12,5/12,7/96,7/96,13/32  

21_11/4     81.94     0,5/6,5/6,1/3,1/3,1/2,9/32,9/32,59/96,59/96,59/96,59/96,1/12,1/12,1/4,3/4,7/12,7/12,29/32,23/96,23/96  

21_19/4     81.94     0,5/6,5/6,1/3,1/3,1/2,83/96,83/96,83/96,83/96,17/32,17/32,1/12,1/12,1/4,3/4,7/12,7/12,5/32,47/96,47/96  

21_19/4     81.94     0,1/6,1/6,2/3,2/3,1/2,1/32,1/32,67/96,67/96,67/96,67/96,11/12,11/12,1/4,3/4,5/12,5/12,31/96,31/96,21/32  

21_15/4     81.94     0,5/6,5/6,1/3,1/3,1/2,71/96,71/96,71/96,71/96,13/32,13/32,1/12,1/12,1/4,3/4,7/12,7/12,1/32,35/96,35/96  

21_15/4     81.94     0,1/6,1/6,2/3,2/3,1/2,29/32,29/32,55/96,55/96,55/96,55/96,11/12,11/12,1/4,3/4,5/12,5/12,19/96,19/96,17/32  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_2     28     0,0,1/7,2/7,4/7,1/8,5/8 1,1,2,2,2,√(7),√(7) 

7_2     28     0,0,1/7,2/7,4/7,7/8,3/8  

14_1     56     0,0,3/4,3/4,1/28,25/28,1/7,2/7,9/28,4/7,1/8,7/8,3/8,5/8  

14_1     56     0,0,3/4,3/4,1/28,25/28,1/7,2/7,9/28,4/7,1/16,1/16,9/16,9/16  

14_1     56     0,0,3/4,3/4,1/28,25/28,1/7,2/7,9/28,4/7,13/16,13/16,5/16,5/16  

14_3     56     0,0,1/4,1/4,1/7,23/28,2/7,11/28,4/7,15/28,1/8,7/8,3/8,5/8  

14_3     56     0,0,1/4,1/4,1/7,23/28,2/7,11/28,4/7,15/28,3/16,3/16,11/16,11/16  

14_3     56     0,0,1/4,1/4,1/7,23/28,2/7,11/28,4/7,15/28,15/16,15/16,7/16,7/16  

21_0     84     0,0,2/3,2/3,2/3,2/3,20/21,20/21,1/7,17/21,17/21,5/21,5/21,2/7,4/7,1/24,1/24,7/8,3/8,13/24,13/24  

21_0     84     0,0,2/3,2/3,2/3,2/3,20/21,20/21,1/7,17/21,17/21,5/21,5/21,2/7,4/7,1/8,19/24,19/24,7/24,7/24,5/8  

21_4     84     0,0,1/3,1/3,1/3,1/3,19/21,19/21,1/7,2/7,13/21,13/21,4/7,10/21,10/21,23/24,23/24,1/8,5/8,11/24,11/24  

21_4     84     0,0,1/3,1/3,1/3,1/3,19/21,19/21,1/7,2/7,13/21,13/21,4/7,10/21,10/21,7/8,5/24,5/24,17/24,17/24,3/8  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_6     28     0,0,6/7,5/7,3/7,7/8,3/8 1,1,2,2,2,√(7),√(7) 

7_6     28     0,0,6/7,5/7,3/7,1/8,5/8  

14_7     56     0,0,1/4,1/4,27/28,3/28,6/7,5/7,19/28,3/7,1/8,7/8,3/8,5/8  

14_7     56     0,0,1/4,1/4,27/28,3/28,6/7,5/7,19/28,3/7,15/16,15/16,7/16,7/16  

14_7     56     0,0,1/4,1/4,27/28,3/28,6/7,5/7,19/28,3/7,3/16,3/16,11/16,11/16  

14_5     56     0,0,3/4,3/4,6/7,5/28,5/7,17/28,3/7,13/28,1/8,7/8,3/8,5/8  

14_5     56     0,0,3/4,3/4,6/7,5/28,5/7,17/28,3/7,13/28,13/16,13/16,5/16,5/16  

14_5     56     0,0,3/4,3/4,6/7,5/28,5/7,17/28,3/7,13/28,1/16,1/16,9/16,9/16  

21_0     84     0,0,1/3,1/3,1/3,1/3,1/21,1/21,6/7,4/21,4/21,16/21,16/21,5/7,3/7,7/8,5/24,5/24,17/24,17/24,3/8  

21_0     84     0,0,1/3,1/3,1/3,1/3,1/21,1/21,6/7,4/21,4/21,16/21,16/21,5/7,3/7,23/24,23/24,1/8,5/8,11/24,11/24  

21_4     84     0,0,2/3,2/3,2/3,2/3,2/21,2/21,6/7,5/7,8/21,8/21,3/7,11/21,11/21,1/8,19/24,19/24,7/24,7/24,5/8  

21_4     84     0,0,2/3,2/3,2/3,2/3,2/21,2/21,6/7,5/7,8/21,8/21,3/7,11/21,11/21,1/24,1/24,7/8,3/8,13/24,13/24  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_8/5     86.75     0,4/5,2/15,0,2/5,1/3,4/5 1,ζ_13^1,ζ_13^2,ζ_13^3,ζ_13^4,ζ_13^5,ζ_13^6 

14_3/5     173.5     0,3/4,4/5,11/20,53/60,2/15,0,3/4,3/20,2/5,1/12,1/3,4/5,11/20  

14_13/5     173.5     0,1/4,1/20,4/5,2/15,23/60,0,1/4,13/20,2/5,1/3,7/12,1/20,4/5  

21_38/5     260.2     0,2/3,2/3,4/5,7/15,7/15,2/15,4/5,4/5,0,2/3,2/3,1/15,1/15,2/5,0,0,1/3,4/5,7/15,7/15  

21_18/5     260.2     0,1/3,1/3,2/15,2/15,4/5,2/15,7/15,7/15,0,1/3,1/3,11/15,11/15,2/5,1/3,2/3,2/3,2/15,2/15,4/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_32/5     86.75     0,1/5,13/15,0,3/5,2/3,1/5 1,ζ_13^1,ζ_13^2,ζ_13^3,ζ_13^4,ζ_13^5,ζ_13^6 

14_37/5     173.5     0,1/4,1/5,9/20,7/60,13/15,0,1/4,17/20,3/5,11/12,2/3,1/5,9/20  

14_27/5     173.5     0,3/4,19/20,1/5,13/15,37/60,0,3/4,7/20,3/5,2/3,5/12,19/20,1/5  

21_2/5     260.2     0,1/3,1/3,1/5,8/15,8/15,13/15,1/5,1/5,0,1/3,1/3,14/15,14/15,3/5,0,0,2/3,1/5,8/15,8/15  

21_22/5     260.2     0,2/3,2/3,13/15,13/15,1/5,13/15,8/15,8/15,0,2/3,2/3,4/15,4/15,3/5,1/3,1/3,2/3,13/15,13/15,1/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_1     93.25     0,1/2,1/2,1/4,1/4,5/8,0 1,ζ_6^2,ζ_6^2,2+√(2),2+√(2),2ζ_6^2,3+√(8) 

14_0     186.5     0,3/4,1/4,1/4,1/2,1/2,0,0,1/4,1/4,3/8,5/8,0,3/4  

14_2     186.5     0,1/4,3/4,3/4,1/2,1/2,1/4,1/4,1/2,1/2,7/8,5/8,0,1/4  

21_7     279.7     0,2/3,2/3,1/6,1/6,1/6,1/6,1/2,1/2,11/12,11/12,11/12,11/12,1/4,1/4,7/24,7/24,5/8,0,2/3,2/3  

21_3     279.7     0,1/3,1/3,5/6,5/6,5/6,5/6,1/2,1/2,1/4,1/4,7/12,7/12,7/12,7/12,23/24,23/24,5/8,0,1/3,1/3  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
7_7     93.25     0,1/2,1/2,3/4,3/4,3/8,0 1,ζ_6^2,ζ_6^2,2+√(2),2+√(2),2ζ_6^2,3+√(8) 

14_0     186.5     0,1/4,3/4,3/4,1/2,1/2,0,0,3/4,3/4,3/8,5/8,0,1/4  

14_6     186.5     0,3/4,1/4,1/4,1/2,1/2,3/4,3/4,1/2,1/2,1/8,3/8,0,3/4  

21_1     279.7     0,1/3,1/3,5/6,5/6,5/6,5/6,1/2,1/2,1/12,1/12,1/12,1/12,3/4,3/4,17/24,17/24,3/8,0,1/3,1/3  

21_5     279.7     0,2/3,2/3,1/6,1/6,1/6,1/6,1/2,1/2,3/4,3/4,5/12,5/12,5/12,5/12,1/24,1/24,3/8,0,2/3,2/3  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_2/5     47.36     0,2/5,2/5,2/5,4/5,4/5,4/5,1/5 1,ζ_3^1,ζ_3^1,ζ_3^1,ζ_8^2,ζ_8^2,ζ_8^2,2+√(5) 

16_37/5     94.72     0,3/4,3/20,3/20,3/20,2/5,2/5,2/5,4/5,4/5,4/5,11/20,11/20,11/20,19/20,1/5  

16_7/5     94.72     0,1/4,13/20,13/20,13/20,2/5,2/5,2/5,1/20,1/20,1/20,4/5,4/5,4/5,1/5,9/20  

24_32/5     142.0     0,2/3,2/3,1/15,1/15,1/15,1/15,1/15,1/15,2/5,2/5,2/5,4/5,4/5,4/5,7/15,7/15,7/15,7/15,7/15,7/15,13/15,13/15,1/5  

24_12/5     142.0     0,1/3,1/3,11/15,11/15,11/15,11/15,11/15,11/15,2/5,2/5,2/5,2/15,2/15,2/15,2/15,2/15,2/15,4/5,4/5,4/5,1/5,8/15,8/15  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_14/5     47.36     0,2/5,2/5,3/5,0,0,4/5,2/5 1,ζ_3^1,ζ_3^1,ζ_3^1,ζ_8^2,ζ_8^2,ζ_8^2,2+√(5) 

16_9/5     94.72     0,3/4,3/20,3/20,7/20,2/5,2/5,3/5,0,0,4/5,3/4,3/4,11/20,3/20,2/5  

16_19/5     94.72     0,1/4,17/20,13/20,13/20,2/5,2/5,3/5,0,0,1/20,4/5,1/4,1/4,13/20,2/5  

24_4/5     142.0     0,2/3,2/3,1/15,1/15,1/15,1/15,4/15,4/15,2/5,2/5,3/5,0,0,4/5,2/3,2/3,2/3,2/3,7/15,7/15,1/15,1/15,2/5  

24_24/5     142.0     0,1/3,1/3,14/15,14/15,11/15,11/15,11/15,11/15,2/5,2/5,3/5,0,0,2/15,2/15,4/5,1/3,1/3,1/3,1/3,11/15,11/15,2/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_26/5     47.36     0,2/5,3/5,3/5,0,0,1/5,3/5 1,ζ_3^1,ζ_3^1,ζ_3^1,ζ_8^2,ζ_8^2,ζ_8^2,2+√(5) 

16_31/5     94.72     0,1/4,17/20,17/20,13/20,2/5,3/5,3/5,0,0,1/5,1/4,1/4,9/20,17/20,3/5  

16_21/5     94.72     0,3/4,3/20,7/20,7/20,2/5,3/5,3/5,0,0,19/20,1/5,3/4,3/4,7/20,3/5  

24_36/5     142.0     0,1/3,1/3,14/15,14/15,14/15,14/15,11/15,11/15,2/5,3/5,3/5,0,0,1/5,1/3,1/3,1/3,1/3,8/15,8/15,14/15,14/15,3/5  

24_16/5     142.0     0,2/3,2/3,1/15,1/15,4/15,4/15,4/15,4/15,2/5,3/5,3/5,0,0,13/15,13/15,1/5,2/3,2/3,2/3,2/3,4/15,4/15,3/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_38/5     47.36     0,3/5,3/5,3/5,1/5,1/5,1/5,4/5 1,ζ_3^1,ζ_3^1,ζ_3^1,ζ_8^2,ζ_8^2,ζ_8^2,2+√(5) 

16_3/5     94.72     0,1/4,17/20,17/20,17/20,3/5,3/5,3/5,1/5,1/5,1/5,9/20,9/20,9/20,1/20,4/5  

16_33/5     94.72     0,3/4,7/20,7/20,7/20,3/5,3/5,3/5,19/20,19/20,19/20,1/5,1/5,1/5,4/5,11/20  

24_8/5     142.0     0,1/3,1/3,14/15,14/15,14/15,14/15,14/15,14/15,3/5,3/5,3/5,1/5,1/5,1/5,8/15,8/15,8/15,8/15,8/15,8/15,2/15,2/15,4/5  

24_28/5     142.0     0,2/3,2/3,4/15,4/15,4/15,4/15,4/15,4/15,3/5,3/5,3/5,13/15,13/15,13/15,13/15,13/15,13/15,1/5,1/5,1/5,4/5,7/15,7/15  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_8/15     69.59     0,3/5,1/3,2/9,2/3,14/15,37/45,4/15 1,ζ_3^1,ζ_7^1,ζ_7^2,ζ_7^3,3.040,4.097,4.658 

16_113/15     139.1     0,3/4,7/20,3/5,1/12,1/3,35/36,2/9,2/3,5/12,14/15,41/60,37/45,103/180,1/60,4/15  

16_23/15     139.1     0,1/4,17/20,3/5,1/3,7/12,2/9,17/36,11/12,2/3,14/15,11/60,13/180,37/45,4/15,31/60  

24_98/15     208.7     0,2/3,2/3,4/15,4/15,3/5,0,0,1/3,8/9,8/9,2/9,1/3,1/3,2/3,14/15,3/5,3/5,37/45,22/45,22/45,14/15,14/15,4/15  

24_38/15     208.7     0,1/3,1/3,14/15,14/15,3/5,1/3,2/3,2/3,2/9,5/9,5/9,0,0,2/3,14/15,4/15,4/15,7/45,7/45,37/45,4/15,3/5,3/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_28/15     69.59     0,3/5,2/3,7/9,1/3,4/15,17/45,14/15 1,ζ_3^1,ζ_7^1,ζ_7^2,ζ_7^3,3.040,4.097,4.658 

16_13/15     139.1     0,3/4,7/20,3/5,2/3,5/12,7/9,19/36,1/12,1/3,1/60,4/15,23/180,17/45,14/15,41/60  

16_43/15     139.1     0,1/4,17/20,3/5,11/12,2/3,1/36,7/9,1/3,7/12,4/15,31/60,113/180,17/45,14/15,11/60  

24_118/15     208.7     0,2/3,2/3,4/15,4/15,3/5,1/3,1/3,2/3,7/9,4/9,4/9,0,0,1/3,14/15,14/15,4/15,2/45,2/45,17/45,14/15,3/5,3/5  

24_58/15     208.7     0,1/3,1/3,14/15,14/15,3/5,0,0,2/3,1/9,1/9,7/9,1/3,2/3,2/3,4/15,3/5,3/5,32/45,32/45,17/45,14/15,4/15,4/15  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_92/15     69.59     0,2/5,1/3,2/9,2/3,11/15,28/45,1/15 1,ζ_3^1,ζ_7^1,ζ_7^2,ζ_7^3,3.040,4.097,4.658 

16_107/15     139.1     0,1/4,13/20,2/5,1/3,7/12,2/9,17/36,11/12,2/3,59/60,11/15,157/180,28/45,1/15,19/60  

16_77/15     139.1     0,3/4,3/20,2/5,1/12,1/3,35/36,2/9,2/3,5/12,11/15,29/60,67/180,28/45,1/15,49/60  

24_2/15     208.7     0,1/3,1/3,11/15,11/15,2/5,1/3,2/3,2/3,2/9,5/9,5/9,0,0,2/3,1/15,1/15,11/15,43/45,43/45,28/45,1/15,2/5,2/5  

24_62/15     208.7     0,2/3,2/3,1/15,1/15,2/5,0,0,1/3,8/9,8/9,2/9,1/3,1/3,2/3,11/15,2/5,2/5,13/45,13/45,28/45,1/15,11/15,11/15  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_112/15     69.59     0,2/5,2/3,7/9,1/3,1/15,8/45,11/15 1,ζ_3^1,ζ_7^1,ζ_7^2,ζ_7^3,3.040,4.097,4.658 

16_7/15     139.1     0,1/4,13/20,2/5,11/12,2/3,1/36,7/9,1/3,7/12,1/15,19/60,8/45,77/180,59/60,11/15  

16_97/15     139.1     0,3/4,3/20,2/5,2/3,5/12,7/9,19/36,1/12,1/3,1/15,49/60,167/180,8/45,11/15,29/60  

24_22/15     208.7     0,1/3,1/3,11/15,11/15,2/5,0,0,2/3,1/9,1/9,7/9,1/3,2/3,2/3,1/15,2/5,2/5,8/45,23/45,23/45,1/15,1/15,11/15  

24_82/15     208.7     0,2/3,2/3,1/15,1/15,2/5,1/3,1/3,2/3,7/9,4/9,4/9,0,0,1/3,1/15,11/15,11/15,38/45,38/45,8/45,11/15,2/5,2/5  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_62/17     125.8     0,5/17,2/17,8/17,6/17,13/17,12/17,3/17 1,ζ_15^1,ζ_15^2,ζ_15^3,ζ_15^4,ζ_15^5,ζ_15^6,ζ_15^7 

16_45/17     251.7     0,3/4,3/68,5/17,2/17,59/68,15/68,8/17,7/68,6/17,13/17,35/68,12/17,31/68,63/68,3/17  

16_79/17     251.7     0,1/4,5/17,37/68,2/17,25/68,49/68,8/17,6/17,41/68,1/68,13/17,65/68,12/17,3/17,29/68  

24_28/17     377.6     0,2/3,2/3,49/51,49/51,5/17,2/17,40/51,40/51,7/51,7/51,8/17,1/51,1/51,6/17,13/17,22/51,22/51,12/17,19/51,19/51,43/51,43/51,3/17  

24_96/17     377.6     0,1/3,1/3,5/17,32/51,32/51,2/17,23/51,23/51,41/51,41/51,8/17,35/51,35/51,6/17,5/51,5/51,13/17,2/51,2/51,12/17,3/17,26/51,26/51  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
8_74/17     125.8     0,12/17,15/17,9/17,11/17,4/17,5/17,14/17 1,ζ_15^1,ζ_15^2,ζ_15^3,ζ_15^4,ζ_15^5,ζ_15^6,ζ_15^7 

16_91/17     251.7     0,1/4,65/68,12/17,15/17,9/68,53/68,9/17,61/68,11/17,4/17,33/68,5/17,37/68,5/68,14/17  

16_57/17     251.7     0,3/4,12/17,31/68,15/17,43/68,19/68,9/17,11/17,27/68,67/68,4/17,3/68,5/17,14/17,39/68  

24_108/17     377.6     0,1/3,1/3,2/51,2/51,12/17,15/17,11/51,11/51,44/51,44/51,9/17,50/51,50/51,11/17,4/17,29/51,29/51,5/17,32/51,32/51,8/51,8/51,14/17  

24_40/17     377.6     0,2/3,2/3,12/17,19/51,19/51,15/17,28/51,28/51,10/51,10/51,9/17,16/51,16/51,11/17,46/51,46/51,4/17,49/51,49/51,5/17,14/17,25/51,25/51  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
9_0     16     0,0,1/2,1/2,1/16,15/16,7/16,9/16,0 1,1,1,1,ζ_2^1,ζ_2^1,ζ_2^1,ζ_2^1,2 

9_0     16     0,0,1/2,1/2,3/16,13/16,5/16,11/16,0  

9_1     16     0,0,1/2,1/2,1/16,1/16,9/16,9/16,1/8  

9_1     16     0,0,1/2,1/2,15/16,3/16,11/16,7/16,1/8  

9_1     16     0,0,1/2,1/2,13/16,13/16,5/16,5/16,1/8  

9_7     16     0,0,1/2,1/2,3/16,3/16,11/16,11/16,7/8  

9_7     16     0,0,1/2,1/2,1/16,13/16,5/16,9/16,7/8  

9_7     16     0,0,1/2,1/2,15/16,15/16,7/16,7/16,7/8  

9_2     16     0,0,1/2,1/2,1/16,3/16,11/16,9/16,1/4  

9_2     16     0,0,1/2,1/2,15/16,13/16,5/16,7/16,1/4  

9_6     16     0,0,1/2,1/2,15/16,13/16,5/16,7/16,3/4  

9_6     16     0,0,1/2,1/2,1/16,3/16,11/16,9/16,3/4  

9_3     16     0,0,1/2,1/2,3/16,3/16,11/16,11/16,3/8  

9_3     16     0,0,1/2,1/2,1/16,13/16,5/16,9/16,3/8  

9_3     16     0,0,1/2,1/2,15/16,15/16,7/16,7/16,3/8  

9_5     16     0,0,1/2,1/2,13/16,13/16,5/16,5/16,5/8  

9_5     16     0,0,1/2,1/2,15/16,3/16,11/16,7/16,5/8  

9_5     16     0,0,1/2,1/2,1/16,1/16,9/16,9/16,5/8  

9_4     16     0,0,1/2,1/2,3/16,13/16,5/16,11/16,1/2  

9_4     16     0,0,1/2,1/2,1/16,15/16,7/16,9/16,1/2  

18_0     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,1/16,13/16,13/16,5/16,5/16,9/16,9/16,1/8,7/8  

18_0     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,0,0,1/2,1/2,1/2,1/2,1/8,7/8  

18_0     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,7/8,7/8,3/8,3/8,5/8,5/8,1/8,7/8  

18_0     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/4,1/4,1/4,1/4,3/4,3/4,3/4,3/4,1/8,7/8  

18_1     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,15/16,3/16,13/16,5/16,11/16,7/16,9/16,0,1/4  

18_1     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,1/8,1/8,5/8,5/8,1/2,1/2,0,1/4  

18_1     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,7/8,7/8,1/4,1/4,3/4,3/4,3/8,3/8,0,1/4  

18_7     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,15/16,3/16,13/16,5/16,11/16,7/16,9/16,0,3/4  

18_7     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,1/4,1/4,3/4,3/4,5/8,5/8,0,3/4  

18_7     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,7/8,7/8,3/8,3/8,1/2,1/2,0,3/4  

18_2     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,1/16,13/16,13/16,5/16,5/16,9/16,9/16,1/8,3/8  

18_2     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,1/8,1/8,5/8,5/8,5/8,5/8,1/8,3/8  

18_2     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,3/8  

18_2     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,7/8,7/8,7/8,7/8,3/8,3/8,3/8,3/8,1/8,3/8  

18_6     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,1/16,13/16,13/16,5/16,5/16,9/16,9/16,7/8,5/8  

18_6     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,7/8,7/8,7/8,7/8,3/8,3/8,3/8,3/8,7/8,5/8  

18_6     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,1/4,1/4,3/4,3/4,1/2,1/2,7/8,5/8  

18_6     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,1/8,1/8,5/8,5/8,5/8,5/8,7/8,5/8  

18_3     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,15/16,3/16,13/16,5/16,11/16,7/16,9/16,1/4,1/2  

18_3     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,1/4,1/4,3/4,3/4,5/8,5/8,1/4,1/2  

18_3     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,7/8,7/8,3/8,3/8,1/2,1/2,1/4,1/2  

18_5     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,15/16,3/16,13/16,5/16,11/16,7/16,9/16,3/4,1/2  

18_5     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,7/8,7/8,1/4,1/4,3/4,3/4,3/8,3/8,3/4,1/2  

18_5     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,1/8,1/8,5/8,5/8,1/2,1/2,3/4,1/2  

18_4     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/16,1/16,13/16,13/16,5/16,5/16,9/16,9/16,3/8,5/8  

18_4     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/4,1/4,1/4,1/4,3/4,3/4,3/4,3/4,3/8,5/8  

18_4     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,1/8,1/8,7/8,7/8,3/8,3/8,5/8,5/8,3/8,5/8  

18_4     32     0,0,1/4,1/4,3/4,3/4,1/2,1/2,0,0,0,0,1/2,1/2,1/2,1/2,3/8,5/8  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
9_5/14     37.18     0,1/2,3/16,1/7,9/14,3/14,5/7,37/112,101/112 1,1,ζ_2^1,ζ_5^1,ζ_5^1,ζ_5^2,ζ_5^2,2.548,3.177 

9_103/14     37.18     0,1/2,1/16,1/7,9/14,3/14,5/7,23/112,87/112  

9_19/14     37.18     0,1/2,5/16,1/7,9/14,3/14,5/7,51/112,3/112  

9_89/14     37.18     0,1/2,15/16,1/7,9/14,3/14,5/7,9/112,73/112  

9_33/14     37.18     0,1/2,7/16,1/7,9/14,3/14,5/7,65/112,17/112  

9_75/14     37.18     0,1/2,13/16,1/7,9/14,3/14,5/7,107/112,59/112  

9_47/14     37.18     0,1/2,9/16,1/7,9/14,3/14,5/7,79/112,31/112  

9_61/14     37.18     0,1/2,11/16,1/7,9/14,3/14,5/7,93/112,45/112  

18_5/14     74.36     0,1/4,3/4,1/2,1/16,5/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,23/112,51/112,3/112,87/112  

18_103/14     74.36     0,1/4,3/4,1/2,15/16,3/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,9/112,37/112,101/112,73/112  

18_19/14     74.36     0,1/4,3/4,1/2,3/16,7/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,37/112,65/112,101/112,17/112  

18_89/14     74.36     0,1/4,3/4,1/2,1/16,13/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,107/112,23/112,87/112,59/112  

18_33/14     74.36     0,1/4,3/4,1/2,5/16,9/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,79/112,51/112,3/112,31/112  

18_75/14     74.36     0,1/4,3/4,1/2,15/16,11/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,9/112,93/112,73/112,45/112  

18_47/14     74.36     0,1/4,3/4,1/2,11/16,7/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,93/112,65/112,17/112,45/112  

18_61/14     74.36     0,1/4,3/4,1/2,13/16,9/16,25/28,1/7,9/14,11/28,27/28,3/14,5/7,13/28,107/112,79/112,31/112,59/112  

27_5/14     111.5     0,1/6,1/6,2/3,2/3,1/2,5/48,5/48,7/16,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,83/336,83/336,65/112,17/112,275/336,275/336  

27_5/14     111.5     0,5/6,5/6,1/3,1/3,1/2,15/16,13/48,13/48,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,9/112,139/336,139/336,331/336,331/336,73/112  

27_103/14     111.5     0,1/6,1/6,2/3,2/3,1/2,47/48,47/48,5/16,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,41/336,41/336,51/112,3/112,233/336,233/336  

27_103/14     111.5     0,5/6,5/6,1/3,1/3,1/2,7/48,7/48,13/16,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,107/112,97/336,97/336,289/336,289/336,59/112  

27_19/14     111.5     0,1/6,1/6,2/3,2/3,1/2,11/48,11/48,9/16,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,79/112,125/336,125/336,317/336,317/336,31/112  

27_19/14     111.5     0,5/6,5/6,1/3,1/3,1/2,1/16,19/48,19/48,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,23/112,181/336,181/336,37/336,37/336,87/112  

27_89/14     111.5     0,1/6,1/6,2/3,2/3,1/2,41/48,41/48,3/16,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,335/336,335/336,37/112,101/112,191/336,191/336  

27_89/14     111.5     0,5/6,5/6,1/3,1/3,1/2,1/48,1/48,11/16,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,55/336,55/336,93/112,247/336,247/336,45/112  

27_33/14     111.5     0,1/6,1/6,2/3,2/3,1/2,11/16,17/48,17/48,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,93/112,167/336,167/336,23/336,23/336,45/112  

27_33/14     111.5     0,5/6,5/6,1/3,1/3,1/2,3/16,25/48,25/48,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,37/112,223/336,223/336,101/112,79/336,79/336  

27_75/14     111.5     0,1/6,1/6,2/3,2/3,1/2,1/16,35/48,35/48,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,293/336,293/336,23/112,87/112,149/336,149/336  

27_75/14     111.5     0,5/6,5/6,1/3,1/3,1/2,43/48,43/48,9/16,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,13/336,13/336,79/112,31/112,205/336,205/336  

27_47/14     111.5     0,1/6,1/6,2/3,2/3,1/2,13/16,23/48,23/48,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,107/112,209/336,209/336,65/336,65/336,59/112  

27_47/14     111.5     0,5/6,5/6,1/3,1/3,1/2,5/16,31/48,31/48,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,265/336,265/336,51/112,3/112,121/336,121/336  

27_61/14     111.5     0,1/6,1/6,2/3,2/3,1/2,15/16,29/48,29/48,1/7,17/21,17/21,13/42,13/42,9/14,37/42,37/42,3/14,5/7,8/21,8/21,9/112,251/336,251/336,107/336,107/336,73/112  

27_61/14     111.5     0,5/6,5/6,1/3,1/3,1/2,37/48,37/48,7/16,41/42,41/42,1/7,9/14,10/21,10/21,1/21,1/21,3/14,5/7,23/42,23/42,307/336,307/336,65/112,17/112,163/336,163/336  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
9_107/14     37.18     0,1/2,13/16,6/7,5/14,11/14,2/7,75/112,11/112 1,1,ζ_2^1,ζ_5^1,ζ_5^1,ζ_5^2,ζ_5^2,2.548,3.177 

9_9/14     37.18     0,1/2,15/16,6/7,5/14,11/14,2/7,89/112,25/112  

9_93/14     37.18     0,1/2,11/16,6/7,5/14,11/14,2/7,61/112,109/112  

9_23/14     37.18     0,1/2,1/16,6/7,5/14,11/14,2/7,103/112,39/112  

9_79/14     37.18     0,1/2,9/16,6/7,5/14,11/14,2/7,47/112,95/112  

9_37/14     37.18     0,1/2,3/16,6/7,5/14,11/14,2/7,5/112,53/112  

9_65/14     37.18     0,1/2,7/16,6/7,5/14,11/14,2/7,33/112,81/112  

9_51/14     37.18     0,1/2,5/16,6/7,5/14,11/14,2/7,19/112,67/112  

18_107/14     74.36     0,1/4,3/4,1/2,15/16,11/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,89/112,61/112,109/112,25/112  

18_9/14     74.36     0,1/4,3/4,1/2,1/16,13/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,103/112,75/112,11/112,39/112  

18_93/14     74.36     0,1/4,3/4,1/2,13/16,9/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,75/112,47/112,11/112,95/112  

18_23/14     74.36     0,1/4,3/4,1/2,15/16,3/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,5/112,89/112,25/112,53/112  

18_79/14     74.36     0,1/4,3/4,1/2,11/16,7/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,33/112,61/112,109/112,81/112  

18_37/14     74.36     0,1/4,3/4,1/2,1/16,5/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,103/112,19/112,39/112,67/112  

18_65/14     74.36     0,1/4,3/4,1/2,5/16,9/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,19/112,47/112,95/112,67/112  

18_51/14     74.36     0,1/4,3/4,1/2,3/16,7/16,3/28,6/7,5/14,17/28,1/28,11/14,2/7,15/28,5/112,33/112,81/112,53/112  

27_107/14     111.5     0,1/6,1/6,2/3,2/3,1/2,1/16,35/48,35/48,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,103/112,197/336,197/336,5/336,5/336,39/112  

27_107/14     111.5     0,5/6,5/6,1/3,1/3,1/2,43/48,43/48,9/16,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,253/336,253/336,47/112,95/112,61/336,61/336  

27_9/14     111.5     0,1/6,1/6,2/3,2/3,1/2,41/48,41/48,3/16,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,5/112,239/336,239/336,47/336,47/336,53/112  

27_9/14     111.5     0,5/6,5/6,1/3,1/3,1/2,1/48,1/48,11/16,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,295/336,295/336,61/112,109/112,103/336,103/336  

27_93/14     111.5     0,1/6,1/6,2/3,2/3,1/2,15/16,29/48,29/48,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,89/112,155/336,155/336,299/336,299/336,25/112  

27_93/14     111.5     0,5/6,5/6,1/3,1/3,1/2,37/48,37/48,7/16,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,33/112,211/336,211/336,19/336,19/336,81/112  

27_23/14     111.5     0,1/6,1/6,2/3,2/3,1/2,47/48,47/48,5/16,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,281/336,281/336,19/112,89/336,89/336,67/112  

27_23/14     111.5     0,5/6,5/6,1/3,1/3,1/2,7/48,7/48,13/16,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,1/336,1/336,75/112,11/112,145/336,145/336  

27_79/14     111.5     0,1/6,1/6,2/3,2/3,1/2,13/16,23/48,23/48,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,75/112,113/336,113/336,11/112,257/336,257/336  

27_79/14     111.5     0,5/6,5/6,1/3,1/3,1/2,5/16,31/48,31/48,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,19/112,169/336,169/336,313/336,313/336,67/112  

27_37/14     111.5     0,1/6,1/6,2/3,2/3,1/2,5/48,5/48,7/16,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,323/336,323/336,33/112,81/112,131/336,131/336  

27_37/14     111.5     0,5/6,5/6,1/3,1/3,1/2,15/16,13/48,13/48,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,43/336,43/336,89/112,25/112,187/336,187/336  

27_65/14     111.5     0,1/6,1/6,2/3,2/3,1/2,11/16,17/48,17/48,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,71/336,71/336,61/112,109/112,215/336,215/336  

27_65/14     111.5     0,5/6,5/6,1/3,1/3,1/2,3/16,25/48,25/48,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,5/112,127/336,127/336,271/336,271/336,53/112  

27_51/14     111.5     0,1/6,1/6,2/3,2/3,1/2,11/48,11/48,9/16,1/42,1/42,6/7,5/14,11/21,11/21,20/21,20/21,11/14,2/7,19/42,19/42,29/336,29/336,47/112,95/112,173/336,173/336  

27_51/14     111.5     0,5/6,5/6,1/3,1/3,1/2,1/16,19/48,19/48,6/7,4/21,4/21,29/42,29/42,5/14,5/42,5/42,11/14,2/7,13/21,13/21,103/112,85/336,85/336,229/336,229/336,39/112  




 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
9_12/5     52.36     0,0,33/40,13/40,1/5,1/5,1/8,5/8,3/5 1,1,ζ_8^1,ζ_8^1,ζ_8^2,ζ_8^2,ζ_8^3,ζ_8^3,ζ_8^4 

9_12/5     52.36     0,0,3/40,23/40,1/5,1/5,7/8,3/8,3/5  

18_7/5     104.7     0,0,3/4,3/4,3/40,33/40,13/40,23/40,19/20,19/20,1/5,1/5,1/8,7/8,3/8,5/8,7/20,3/5  

18_7/5     104.7     0,0,3/4,3/4,61/80,61/80,21/80,21/80,19/20,19/20,1/5,1/5,1/16,1/16,9/16,9/16,7/20,3/5  

18_7/5     104.7     0,0,3/4,3/4,1/80,1/80,41/80,41/80,19/20,19/20,1/5,1/5,13/16,13/16,5/16,5/16,7/20,3/5  

18_17/5     104.7     0,0,1/4,1/4,3/40,33/40,13/40,23/40,1/5,1/5,9/20,9/20,1/8,7/8,3/8,5/8,17/20,3/5  

18_17/5     104.7     0,0,1/4,1/4,11/80,11/80,51/80,51/80,1/5,1/5,9/20,9/20,15/16,15/16,7/16,7/16,17/20,3/5  

18_17/5     104.7     0,0,1/4,1/4,71/80,71/80,31/80,31/80,1/5,1/5,9/20,9/20,3/16,3/16,11/16,11/16,17/20,3/5  

27_2/5     157.0     0,0,2/3,2/3,2/3,2/3,3/40,29/120,29/120,89/120,89/120,23/40,13/15,13/15,13/15,13/15,1/5,1/5,1/24,1/24,7/8,3/8,13/24,13/24,4/15,4/15,3/5  

27_2/5     157.0     0,0,2/3,2/3,2/3,2/3,119/120,119/120,33/40,13/40,59/120,59/120,13/15,13/15,13/15,13/15,1/5,1/5,1/8,19/24,19/24,7/24,7/24,5/8,4/15,4/15,3/5  

27_22/5     157.0     0,0,1/3,1/3,1/3,1/3,19/120,19/120,33/40,13/40,79/120,79/120,1/5,1/5,8/15,8/15,8/15,8/15,23/24,23/24,1/8,5/8,11/24,11/24,14/15,14/15,3/5  

27_22/5     157.0     0,0,1/3,1/3,1/3,1/3,3/40,109/120,109/120,49/120,49/120,23/40,1/5,1/5,8/15,8/15,8/15,8/15,7/8,5/24,5/24,17/24,17/24,3/8,14/15,14/15,3/5  




 

 


1.25 

N_c     D^2     s_1,s_2,⋯ d_1,d_2,⋯ 
 
9_28/5     52.36     0,0,7/40,27/40,4/5,4/5,7/8,3/8,2/5 1,1,ζ_8^1,ζ_8^1,ζ_8^2,ζ_8^2,ζ_8^3,ζ_8^3,ζ_8^4 

9_28/5     52.36     0,0,37/40,17/40,4/5,4/5,1/8,5/8,2/5  

18_33/5     104.7     0,0,1/4,1/4,37/40,7/40,27/40,17/40,1/20,1/20,4/5,4/5,1/8,7/8,3/8,5/8,13/20,2/5  

18_33/5     104.7     0,0,1/4,1/4,19/80,19/80,59/80,59/80,1/20,1/20,4/5,4/5,15/16,15/16,7/16,7/16,13/20,2/5  

18_33/5     104.7     0,0,1/4,1/4,79/80,79/80,39/80,39/80,1/20,1/20,4/5,4/5,3/16,3/16,11/16,11/16,13/20,2/5  

18_23/5     104.7     0,0,3/4,3/4,37/40,7/40,27/40,17/40,4/5,4/5,11/20,11/20,1/8,7/8,3/8,5/8,3/20,2/5  

18_23/5     104.7     0,0,3/4,3/4,9/80,9/80,49/80,49/80,4/5,4/5,11/20,11/20,13/16,13/16,5/16,5/16,3/20,2/5  

18_23/5     104.7     0,0,3/4,3/4,69/80,69/80,29/80,29/80,4/5,4/5,11/20,11/20,1/16,1/16,9/16,9/16,3/20,2/5  

27_38/5     157.0     0,0,1/3,1/3,1/3,1/3,37/40,91/120,91/120,31/120,31/120,17/40,2/15,2/15,2/15,2/15,4/5,4/5,23/24,23/24,1/8,5/8,11/24,11/24,11/15,11/15,2/5  

27_38/5     157.0     0,0,1/3,1/3,1/3,1/3,1/120,1/120,7/40,27/40,61/120,61/120,2/15,2/15,2/15,2/15,4/5,4/5,7/8,5/24,5/24,17/24,17/24,3/8,11/15,11/15,2/5  

27_18/5     157.0     0,0,2/3,2/3,2/3,2/3,37/40,11/120,11/120,71/120,71/120,17/40,4/5,4/5,7/15,7/15,7/15,7/15,1/8,19/24,19/24,7/24,7/24,5/8,1/15,1/15,2/5  

27_18/5     157.0     0,0,2/3,2/3,2/3,2/3,101/120,101/120,7/40,27/40,41/120,41/120,4/5,4/5,7/15,7/15,7/15,7/15,1/24,1/24,7/8,3/8,13/24,13/24,1/15,1/15,2/5  








