The weighted moduli spaces of sextics
Lubjana Beshaj Scott Guest

	


	

Department of Mathematical Sciences, 

 United States Military Academy, 

 West Point, NY, 10996
lubjana.beshaj@usma.edu
	
	

Department of Mathematics and Statistics,

 Oakland University, 

 Rochester, MI 48309
sguest@oakland.edu
	

We use the weighted moduli height as defined in <cit.> to investigate the distribution of fine moduli points in the moduli space of genus two curves.
 We show that for any genus two curve with equation y^2=f(x),  its weighted moduli height () ≤ 2^3 √(3 · 5 · 7) ·  H(f), where H(f) is the minimal naive height of the curve as defined in <cit.>.
Based on the weighted moduli height  we create a database of genus two curves defined over  with small    and show that for small such height  (< 5) about 30% of points are fine moduli points.

	
[2010]14H10,14H45
	
The weighted moduli spaces of sextics
    Scott Guest
    
=====================================









§ INTRODUCTION


In <cit.> authors studied heights of algebraic curves and compared the naive height with the moduli height.  Their examples focused mostly on curves of genus two.
In  <cit.>, the authors created a database of rational points in the moduli space of genus two curves _2  with moduli height 𝔥≤ 20 and considered the problem of what percentage of these points have corresponding genus 2 curves defined over .    It is widely believed that for  almost all points ∈_2 (), there is no genus two representative defined over . In the language of arithmetic geometry this means that for almost all genus two curves the field of moduli is not a minimal field of definition.
However,  results in <cit.> were somewhat surprising. For all 𝔥≤ 20 roughly 30% of the points the field of moduli was a field of definition. There were a couple interpretations to this.

Firstly, the points in  _2 () were represented using the absolute invariants i_1, i_2, i_3 of genus two curves.   These are rational functions in terms of the Igusa invariants J_2, J_4, J_6, and J_10, see <cit.>, <cit.>, <cit.>, but they are not defined when J_2 = 0.  To handle the case J_2 =0 another compactification of the moduli space was used.  When using two different sets of invariants however, there is no natural means to make statements about interesting questions regarding the moduli space of genus two curves.

Secondly, it is possible that the "majority" of moduli points for which the field of moduli is a field of definition are concentrated around the "center" of the moduli space.  In other words, as the moduli height 𝔥 goes to ∞, the percentage of curves for which the field of moduli is a field of definition tends to zero.  However, computations in <cit.>  didn't quite show this trend.

In <cit.> a "universal" equation of genus two curves was discovered.  Here universal means that it works for any tuple of arithmetic invariants (J_2, J_4, J_6, J_10) including the case when J_2=0 or the case when the automorphism group has order > 2, which was treated separately in Mestre's original algorithm <cit.> and has not been implemented in most computational algebra packages. This raised the question on whether we could perform computations done in <cit.>, but now using [J_2, J_4, J_6, J_10] and the weighted projective space 𝕎ℙ_(2,4,6,10)^3 instead of the coordinate (i_1, i_2, i_3) used in <cit.>?


The main question became whether there was any measure of height for the weighted projective spaces which could be used for 𝕎ℙ_(2,4,6,10)^3 in order to perform a similar analysis as in <cit.>. Luckily, <cit.> came out which does precisely what we needed, defines a height on weighted projective spaces. Using this height for 𝕎ℙ_(2,4,6,10)^3 we were able to determine a unique tuple (J_2, J_4, J_6, J_10) for any point ∈𝕎ℙ_(2,4,6,10)^3 and compile a database of genus two curves based on this weighted projective height.

In this database every point is represented by a unique minimal tuple of arithmetic invariants (J_2, J_4, J_6, J_10).  Such database is valid for curves over any field of characteristic different from two.  In the case of characteristic two we can do exactly the same thing by adding another invariant J_8 as explained in <cit.>.
The disadvantage of using the weighted moduli height is that since powers of invariants are involved, one can not perform computations for very big heights. In this paper we only go for heights up to 𝔥 =4.

This paper is organized as follows.  In <ref>, we start with giving a brief definition of a weighted projective space and then  define the height  function on this space, following closely the approach in  <cit.>.
	
In	<ref>  we explain how we build the database. We start with creating a database of normalized tuples = [J_2: J_4: J_6: J_10] ∈𝕎ℙ_(2,4,6,10)^3() with weighted height ≤ 5 and J_10≠ 0.  Then,  in subsection	<ref> we explain how  for each point ∈_2 () we can compute the equation of the curve over the rationals, when such exists. In <cit.> the authors do exactly this and construct for every point ∈_2 the equation of the corresponding genus 2 curve. While we know that each  is defined over  if it corresponds to a curve defined over , the converse in general is not true. It is then necessary to determine if a given moduli point actually corresponds to a curve over .    We say a moduli point ∈_2 () is a fine  point if and only we can find a representative curve _ defined over , otherwise we call it a coarse point or an obstruction point.
	
For each such tuple = [J_2: J_4: J_6: J_10] ∈𝕎ℙ_(2,4,6,10)^3() ∖{ J_10 = 0} , we also compute the automorphism group of the corresponding curve, using  algorithms presented in <cit.>. We then present the  data on the distribution of fine points in the moduli space of genus two curves. Namely, we compare the number of moduli points below a given weighted moduli height to the number of fine points below that height for all weighted moduli heights < 5.  Finally, we give a table of specific results for all moduli points with < 1.




Notation:  Throughout this paper by a curve we mean a smooth, irreducible algebraic curve. Unless otherwise noted a curve  means the isomorphism class of  over some field k. With 𝕎ℙ_(2, 4, 6, 10)^3 ()  we denote the weighted projective space with weights (2, 4, 6, 10) and  () we denote  the weighted moduli height of a point ∈ 𝕎ℙ_(2, 4, 6, 10)^3 () which we define in the preliminaries.





§ PRELIMINARIES


Let  be a genus 2 curve defined over a field k with characteristic 0 or a prime p ≠ 2. It is a well known fact that the  equation of  can  be given by


    :  y^2    =  f(x) = a_6 ∑_i=1^6 (x - _i) =   a_6 x^6 +  a_5 x^5 z+ ⋯ + a_0 z^6


where {_i}_i=1^6 are the ramification points of the map →^1.   The isomorphism classes of genus 2 curves are on one to one correspondence with the orbits of the GL_2 (k)-action on the space of binary sextics.  Igusa proved that the invariant ring _6 is generated by the Igusa arithmetic invariants J_2, J_4, J_6, J_10; see   <cit.> for details.  Note that Igusa J-invariants {J_2i}, for  i=1, 2, 3, 5, are homogenous polynomials of degree 2i in   k[a_0, …, a_6] and J_10 is the discriminant of the curve.  Since in this paper we are working with smooth, irreducible curves than we take J_10≠ 0.  The definitions of these invariants can be found in several papers such as  <cit.>.  









 §.§ The weighted projective space

	

In this subsection we will follow closely  the notation in <cit.>.     Let k be a field of characteristic zero and  (q_0, … , q_n) a fixed tuple of positive integers called weights.  Consider the action of k^⋆ on ^n+1 as follows


    λ * (x_0, … , x_n) = ( ł^q_0 x_0, … , ł^q_n x_n   )


for ł∈ k^∗.
The quotient of this action is called a weighted projective space and denoted by   𝕎ℙ^n_(q_0, … , q_n).
It is the projective variety Proj ( k [x_0,...,x_n] ) associated to the graded ring k [x_0, … ,x_n] where the variable x_i has degree q_i for i=0, … , n.




In <cit.> they define the concept of height on the weighted projective spaces. In this subsection we will mostly focus on the space 𝕎ℙ_(2,4,6,10)^3()∖{J_10 = 0 }.  Let  = [J_2, J_4, J_6, J_10] ∈𝕎ℙ_(2,4,6,10)^3()∖{J_10 = 0 } be a given weighted moduli  point in the weighted projective space. We define the tuple  to be  a normalized weighted moduli point if the weighted greatest common divisor of its coordinates is one, where the weighted greatest common divisor  is defined as follows


    (J_2i )  = ∏_ v ∈ M_K∏_d^q_i | J_i
     d ∈  |d|_v


for all valuation v in the set of valuations M_K and divisors d ∈ such that for all i=2,4, 6, 10 we have d^i | J_i.  Note that if K = then,


    (J_2i )  =    ∏_p^q_i | J_i
     p ∈  |p|


for all primes p ∈. The normalization of the point ∈𝕎ℙ^3_(K) is


    𝔮 = 1/ ()⋆.


and from   <cit.> we have that since w= (2, 4, 6, 10)   and d = ( 2, 4, 6, 10)=2, then   this normalization is unique up to a multiplication by a d-th root of unity.  In other words, for every  normalized point  = [J_2, J_4, J_6, J_10] ∈𝕎ℙ^3_(), there is another normalized point  ^' = [-J_2, J_4, -J_6, -J_10] equivalent to . Moreover,  and ^' are isomorphic over the Gaussian integers, see <cit.> for more details.












Notice that if the first coordinate J_2=0 then the normalization  in the space 𝕎ℙ_(2,4,6,10)^3 () is unique up to a multiplication by a d= (4, 6, 10)-th root of unity.  For example, the pair of points  and ^' in Table <ref> are twists of each other, where ^' = √(-1)·.   In the coming section we will see why these are the only tuples of height h=1 for which this is true.














Next, we will see  when a point ∈𝕎ℙ^3_w() is normalized over .  The  point  is called absolutely normalized, i.e.  normalized over  if () = 1 where


    () =  ∏_ł^q_i | J_i
    ł∈  |ł|


such that for all i= 2, 4, 6, 10, ł^i ∈. We can compute the absolutely normalization of a point in the weighted projective space the same way as we computed the normalization of that point.

Two absolutely normalized tuples are twists of each other if and only if they are isomorphic over the Gaussian integers.


The  weighted multiplicative height of P=( x_0, … , x_n )  ∈^n_w (K) over K is defined as


    _K (P)= 1/() ∏_v ∈ M_Kmax{ |x_0|_v^1/q_0,  … , |x_n|_v^1/q_n}

for all valuations v is the set of valuations   M_K and the absolute weighted multiplicative height  over 

    (P)= 1/() ∏_v ∈ M_Kmax{ |x_0|_v^1/q_0,  … , |x_n|_v^1/q_n}.


Note that if we consider points ∈𝕎ℙ^3_w() then it is clear that   will have a representative over the integers.  With such a
representative for the coordinates of , the non-Archimedean absolute values give no contribution to the height and the only one that matters is the Archimedean absolute value, i.e. just the maximum of the coordinates of the given point.


Next, we show how we can calculate the absolute minimal tuple and absolute minimum height of all  zero-dimensional loci in _2. All such curves are defined over , so we consider their respective weighted moduli points as existing in 𝕎ℙ^3_(). 


Let  be the genus two curve with equation y^2 = x^6 - 1. Its weighted moduli point is


    _1 = [ 240, 1620, 119880, 46656 ] = [ 2^4 · 3 · 5, 2^2 · 3^4 · 5, 2^3 · 3^4 · 5 · 37, 2^6 · 3^6 ].


which is a minimal tuple since there is no p ∈ such that p^2i | J_2i, for i = 1, 2, 3, 5. This tuple has three twists for values of ł = 2, 3, 6. The corresponding twists are
		
			_2    =1/2 ⋆=  [ 120, 405, 14985, 1458 ]       = [ 2^3 ·3 ·5, 3^4 ·5, 3^4 ·5 ·37, 2 ·3^6 ]
			

			_3    = 1/3 ⋆= [ 80, 180, 4440, 192 ]       = [ 2^4 ·5, 2^2 ·3^2 ·5, 2^3 ·3 ·5 ·37, 2^6 ·3 ]
			

			_4    = 1/6 ⋆= [ 40, 45, 555, 6 ]       = [ 2^3 ·5, 3^2 ·5, 3 ·5 ·37, 2 ·3 ]
		
		The heights are
		
    (_1) = 4√(15), (_2) = 2√(30), (_3) = 4√(5), (_4) = 2√(10).

		The absolute minimal tuple is
		
    = [ 40, 45, 555, 6 ] = [ 2^3 · 5, 3^2 · 5, 3 · 5 · 37, 2 · 3 ].

		Hence, the absolute minimum height is
		
    () = 2√(10).

	
	
	
		Let  be the genus two curve with equation y^2 = x^5 - x. Its weighted moduli point is
		
    _1 = [ -40, -80, 320, -256 ] = [-1 · 2^3 · 5, -1 · 2^4 · 5, 2^6 · 5, -1 · 2^8 ],

		which is a minimal tuple since there is no p ∈ such that p^2i | J_2i, for i = 1, 2, 3, 5. This tuple has a sole twist for the value ł = 2. This twist is
		
    _2 = [ -20, -20, 40, -8 ] = [-1 · 2^2 · 5, -1 · 2^2 · 5, 2^3 · 5, -1 · 2^3 ].

		The heights are
		
    (_1) = 2√(10), (_2) = 2√(5).

		The absolute minimal tuple is
		
    = [ -20, -20, 40, -8 ] = [-1 · 2^2 · 5, -1 · 2^2 · 5, 2^3 · 5, -1 · 2^3 ].

		
		Hence, the absolute minimum height is
		
    () = 2√(5).

		
	
	
	
		Let  be the genus two curve with equation y^2 = x^6- x. Its weighted moduli point is
		
    _1 = [ 0, 0, 0, 3125 ] = [ 0, 0 ,0 , 5^5 ].

		which is a minimal tuple since there is no p ∈ such that p^2i | J_2i, for i = 1, 2, 3, 5. This tuple has a sole twist for the value ł = 5. This twist is
		
    _2 = [ 0, 0, 0, 1]

		The heights are
		
    (_1) = √(3125), (_2) = 1

		The absolute minimal tuple is
		
    = [ 0, 0, 0, 1].

		Hence, the absolute minimum height is
		
    () = 1.

	
	
	 	
In <cit.> was defined the moduli  height  of genus g algebraic curves.  Consider  ∈_g, where _g is the moduli space of smooth  irreducible algebraic curves of genus g ≥ 2.  It is known that _g is a quasi projective variety of dimension 3g-3 and is embedded in ^3g-2. The moduli height is  the usual height in the projective space ^3g-2, i.e. if P ∈^3g-2 () then


    H(P)  = max{/  |x_0|,  … , |x_n|  }.


Moreover, for  g = 2  where a moduli point is given by  = [J_2, …, J_10] ∈_2() then


    H()  =  max{/ |J_2|,  … , |J_10|  }.


If we consider ∈𝕎ℙ_w^3 () then the weighted moduli height is


    () = max{ |J_2|^1/2, …, |J_10|^1/10}.



Hence, by definition for a given moduli point  the weighted moduli height is always smaller than the moduli height.  And this makes computing databases of genus two curves based on the weighted moduli height much more convenient.

	



 §.§ Naive height versus weighted moduli height





Let  be a genus two curve with equation given as


    y^2 =  a_6 x^6 + a_5x^5z + ⋯ + a_0 z^6.


The naive height H() of a curve  (resp. polynomial) is the maximum of the absolute value of the  coefficients of the curve (resp. polynomial);  see <cit.> for details.  In <cit.> and <cit.> were used both the naive height and the moduli height to analyze points in _2 ().  Even though   using the weighted moduli height makes more sense in analyzing points in _2 (),  since a weighted moduli point determines uniquely an isomorphism class of the curves, it would be interesting to see how the weighted moduli height relates to the naive height.    In <cit.> it  is given an inequality how the naive height is compared to the  moduli height. In this subsection we capture previous results and also compare the naive height to the weighted moduli height.


 In <cit.>  the naive height of a curve is compared with the moduli height and  showed that  for a genus 2 curve with equation y^2=f(x) the moduli height is bounded as follows


    (f) ≤ 2^28· 3^9 · 5^5 · 7 · 11 · 13 ·  17 · 43  · H(f)^10.



Note that there is a typo in the computations in <cit.> and the above coefficient should be 2^24· 3^7 · 5^4 · 7 · 11 · 13 · 43.  Next,  we consider how the weighted moduli height is related to the naive height of the curve. 


Let  be a genus 2 curve and  its corresponding point in 𝕎ℙ_w^3 ().  Then   the weighted moduli height is bounded as follows


    () ≤ 2^3  √(3 · 5 · 7)· H ()





We assume that the equation of the curve is given as y^2 = f(x). Recall  that each J_2i is a degree 2i polynomial evaluated at f, i.e degree 2i polynomial  given in   k[a_0, ⋯, a_6].  Then, from  Lemma 15 in  <cit.> we have


    H    (J_2i      (f) )≤ c_0 · H    (J_2i)  · H(f)^2i


where c_0 =  d+n  n is the  number of monomials of a degree d  homogenous polynomial in n+1 variables.  Computations of H(J_2i) is done in Maple and we get


    H(J_2)    ≤  2^6· 3  · 5 · 7 · H(f)^2 
    
    H(J_4)    ≤ 2^3· 3^5· 5^2 · 7 · H(f)^4  
    
    H(J_6)    ≤  2^5 · 3^5 · 5  · 7 · 11 · 37 · H(f)^6   
    
    H(J_10)    ≤ 2^9· 3^5 · 5  · 7 · 11 · 13 · H(f)^10


The weighted moduli height of  is computed as follows


    () = max{ H (J_2(f))^1/2, … , H( J_10 (f))^1/10}≤  2^3 √(3 · 5 · 7)· H(f).


This concludes the proof.




Hence,  we get that the height of the given curve  with corresponding moduli point =[J_2: J_4: J_6: J_10] is as follows


    H ()  ≥1/2^3  √(3 · 5 · 7)· ()


 but  we don't know if this lower bound is strict and if it can ever be achieved.   In <cit.> are displayed all genus 2 curves (up to isomorphism) with naive height H =1. In <cit.> there are displayed 230 isomorphism classes of genus two curves with naive height one.  There are 184 genus two curves with naive height one and automorphism group _2, 30 with automorphism group isomorphic to V_4, 11 with automorphism group D_4, two curves with group D_6, and three with automorphism group respectively 10, 24, and 48.


 In an upcoming paper we will  check how the weighted moduli height of those curves in <cit.>  is compared to their naive height and see if this lower bound is sharp for any of those curves.  



	



§ RATIONAL POINTS IN THE WEIGHTED MODULI SPACE




Next we will build a database of all isomorphism classes of genus 2 curves with bounded weighted moduli height.  We  start with creating a database of normalized tuples  in 𝕎ℙ_w^3 ()∖{J_10 =0} of height ≤ h, for a given h.    Then,  we can compute the equation of the curve for a given ∈𝕎ℙ_w^3 ()∖{J_10 =0} using methods explained in the upcoming  subsection <ref>.







Let h be a positive integer.  The number of  points ∈ () ∖{J_10 = 0} of  weighted moduli height ≤ h is given by the following lemma.
	
 For any given positive integer h there are  at most


    2 h^10 (h^2 + 1) (2h^4 +1) (2h^6 +1)
	

normalized points in  () ∖{J_10 = 0}   of  weighted moduli height ≤ h.
	

Let  be a genus two curve and   = [J_2, J_4, J_6, J_10] ∈ () ∖{J_10 = 0} the moduli point representing  of height () = h.   From the definition of weighted moduli height  for every i=1, 2, 3, 5 we have that


    |J_2i | ^1/2i≤ h

Hence, for a given height h, every element J_2i of  satisfies

 
    -h^2i≤ J_2i≤ h^2i for   i = 1, 2, 3, 5.

 
Since,   J_10≠ 0 then there are 2h^10 possible integer values for J_10.  In <cit.> it is proved that for every  normalized point  = [J_2, J_4, J_6, J_10] ∈𝕎ℙ_(2, 4, 6, 10)^3 (), there is another normalized point  ^' = [-J_2, J_4, -J_6, -J_10] equivalent to .  Since, we only want to count equivalence classes then we let J_2 ≥ 0 and therefore there are h^2+1 possible integer values for J_2.  Lastly, for  i = 2, 3 there are (2 h^2i +1)-possible integer values for J_2i.


But, up to this point we didn't count the points ∈𝕎ℙ_(2, 4, 6, 10)^3 () that satisfy the property that there exists a λ∈^⋆ such that


    λ⋆ (J_2,  …, J_10)  =  (λ^2 J_2, …, λ^10 J_10).


But there is not a rigorous mathematical way of counting this points. Hence, the number of normalized points ∈𝕎ℙ_(2, 4, 6, 10)^3 () will be at most


    2 h^10 (h^2 + 1) (2h^4 +1) (2h^6 +1).
	

 This concludes the proof.



 However, for computational reasons it is more convenient to consider the space 𝕎ℙ_(1,2,3,5)^3 () instead of (). In this case the above lemma becomes as follows.

 For any given positive integer h there are  at most


    2 h^5 (h + 1) (2h^2 +1) (2h^3 +1)
	

normalized points in  𝕎ℙ_(1,2,3,5)^3 ()   of  weighted moduli height ≤ h.
	









The proof is similar to the previous lemma.  In Table <ref>  for  weighted moduli height ≤ 4 we display how many possible tuples are there in 𝕎ℙ_(1, 2, 3, 5)^3 ()  and then  in the next column we display how many of this tuples give us isomorphism classes of genus two curves. And this results agree with Lemma <ref>.





Then, in the upcoming  Table <ref>  we display  all points ∈𝕎ℙ_(1,2,3,5)^3 () ∖{J_10 = 0} with weighted moduli heights = 1.







	
	
	
Note that  in Table <ref>  the points 1 to 15 are fine moduli points, i.e. with methods that we will explain in the upcoming subsection we can compute their equations defined over .    We have already seen points 1 to 6 and 17 to 19 and their twists in the previous section in Table <ref>.
	





 §.§ Equations of curves from points in () 


In <cit.> for every point =[J_2 : J_4 : J_6 :J_10] ∈∖{J_10=0} was constructed a genus two curve defined over a field of definition. We recapture that result below:
		


		Let = [J_2, J_4, J_6, J_10] ∈𝕎ℙ^3_() for some number field K and  its ring of integers. There is a genus-two curve corresponding to  as follows:
		
			
  * If J_2 · J_10≠ 0, then there is a genus 2 curve C_(α, β) given by
			
    C_(α, β) : y^2 = ∑_i=0^6 a_i (α, β) x^i

			
			with coefficients given in  <cit.> and a pair (α, β) satisfying
			
    α^2 + ł_6 β^2 σ = γ,

			where ł_6, σ, and γ are given in terms of some ρ and κ, which are themselves determined by . Their equations are given in <cit.>.  Moreover, C_(α, β) is defined over its field of moduli K, i.e. a_i(α, β) ∈ K, i = 0, …, 6, if and only if K-rational α and β exist. For the explicit equation of C_(α, β) we refer the reader directly to <cit.>.
			
			
  * If J_2 = 0 and J_4 · J_6 · J_10≠ 0, then there is a genus 2 curve given by setting ρ = κ≠ 0 in the previous equation.
			
			
  * If J_2 = J_6 = 0 and J_4 · J_10≠ 0, there is only one genus-two curve given by
			
    y^2 =    (4ν + 1)(2ν - 1)x^6 + 2(1 - ν)(4ν + 3)x^5 - 15(1 - ν)x^4 
       + 20(1 - ν)^2x^3 + 5(2ν - 3)(1- ν)^2x^2 + 6(1 - ν)^3x - (1 - ν)^3,

			where
			
    ν = J_4^5/2^23^55^5J_10^2.

			
			
  * If J_2 = J_4 = 0 and J_6 · J_10≠ 0, there is only one genus-two curve given by
			
    y^2 =    5x^6 + 12(1 - μ)x^5 - 15(1 - μ)x^4 - 80(1 - μ)^2x^3 
       + 15(4μ - 7)(1 - μ)^2x^2 - 60(1 - μ)^3x + (4μ - 13)(1 - μ)^3,

			where
			
    μ = J_6^5/2^43^45^5J_10^3 .

			
			
  * If J_2 = J_4 = J_6 = 0 and J_10≠ 0, there is only one genus-two curve given by
			
    y^2 = x^6 - x.

		


Now that we have seen how to compute equations of curves  we turn back to one of our main goals for this paper: Investigate what proportion of points in () are defined over .  The following remark is believed to be true.



	

For almost all  points  ∈_2 (),  the field of moduli is not a field of definition. In other words, the probability that a point ∈_2 () has a representative curve defined over  is zero.



 However, this was contradicted by previous results in <cit.> where absolute invariants i_1, i_2, i_3 were used to identify the isomorphism classes of curves. At first we believed, this  was because such invariants are not everywhere defined and to build the database two different set of invariants were used, see <cit.> for more details.    But,  as we will see below  we get the same contradicting results using points in 𝕎ℙ_(1, 2, 3, 5)^3 () with minimal weighted moduli height.
The following result  is well known by work of Cardona/Quer <cit.> or  Shaska; see <cit.>.



Every genus 2 curve with automorphism group of order > 2 is defined over its field of moduli. Moreover, its equation over its field of moduli
it's  given in <cit.>.





Hence, we are left to check for curves with automorphism group of order two.
In  Table <ref> given below we  display the number of points with height ≤ 5 for each possible group of automorphisms and  the number of points that  are defined over the rationals.  We stop at =5 since then the computations get to big.  As we can see most of the curves are curves with automorphism group _2, as expected.   From the data we notice that the  percentage of the curves defined over the rationals  gets smaller as the weighted moduli height gets higher  and as we can see  approximately 30% of the points are defined over the rationals.  This results agree with previous results from <cit.> and we expect the ratio to converges to zero as the height gets bigger. It would be interesting  to see   for what moduli height  this ratio is zero.













Acknowledgments.  We want to thank Prof. Tony Shaska for sharing his code for the universal equation of genus two curves from <cit.>  and his helpful discussions and suggestions.






amsplain
	

	
