Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -1, 0]); // or

magma: E := EllipticCurve("35a3");

sage: E = EllipticCurve([0, 1, 1, -1, 0]) # or

sage: E = EllipticCurve("35a3")

gp: E = ellinit([0, 1, 1, -1, 0]) \\ or

gp: E = ellinit("35a3")

$$y^2 + y = x^{3} + x^{2} - x$$

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(0, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(0, 0\right)$$, $$\left(0, -1\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$35$$ = $$5 \cdot 7$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-35$$ = $$-1 \cdot 5 \cdot 7$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{262144}{35}$$ = $$-1 \cdot 2^{18} \cdot 5^{-1} \cdot 7^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$6.32620115221$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1$$  = $$1\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$3$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form35.2.a.a

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} - q^{15} + 4q^{16} + 3q^{17} + 2q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 6 $$\Gamma_0(N)$$-optimal: no Manin constant: 3

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

$$L(E,1)$$ ≈ $$0.702911239135$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 ss ordinary nonsplit split 0,5 0 0 1 0,0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 35a consists of 3 curves linked by isogenies of degrees dividing 9.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.140.1 $$\Z/6\Z$$ Not in database
$$\Q(\zeta_{7})^+$$ $$\Z/9\Z$$ 3.3.49.1-875.1-a2
6 6.0.40516875.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.826875.2 $$\Z/9\Z$$ Not in database
6.0.686000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.
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