Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -544326, 154522980]); // or

magma: E := EllipticCurve("3570v2");

sage: E = EllipticCurve([1, 0, 0, -544326, 154522980]) # or

sage: E = EllipticCurve("3570v2")

gp: E = ellinit([1, 0, 0, -544326, 154522980]) \\ or

gp: E = ellinit("3570v2")

$$y^2 + x y = x^{3} - 544326 x + 154522980$$

## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{6}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(428, -214\right)$$, $$\left(156, 8490\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-852, 426\right)$$, $$\left(156, 8490\right)$$, $$\left(156, -8646\right)$$, $$\left(408, 426\right)$$, $$\left(408, -834\right)$$, $$\left(428, -214\right)$$, $$\left(444, 426\right)$$, $$\left(444, -870\right)$$, $$\left(768, 13386\right)$$, $$\left(768, -14154\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$3570$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$770635366502400$$ = $$2^{12} \cdot 3^{12} \cdot 5^{2} \cdot 7^{2} \cdot 17^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{17836145204788591940449}{770635366502400}$$ = $$2^{-12} \cdot 3^{-12} \cdot 5^{-2} \cdot 7^{-2} \cdot 17^{-2} \cdot 73^{3} \cdot 357913^{3}$$ 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: $$0.474363585755$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1152$$  = $$( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$12$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form3570.2.a.w

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 36864 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### 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)

$$L(E,1)$$ ≈ $$3.79490868604$$

## 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$$(j)_{-}$$
$$2$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$3$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

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
$$2$$ Cs
$$3$$ 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 17 split split nonsplit split nonsplit 6 1 0 1 0 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 $$d=$$ 2, 3 and 6.
Its isogeny class 3570v consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-5}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{7}, \sqrt{-17})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{5}, \sqrt{17})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.3384009916875.2 $$\Z/6\Z \times \Z/6\Z$$ 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.