Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -78500067, -226972151659]); // or

magma: E := EllipticCurve("286650ba2");

sage: E = EllipticCurve([1, -1, 0, -78500067, -226972151659]) # or

sage: E = EllipticCurve("286650ba2")

gp: E = ellinit([1, -1, 0, -78500067, -226972151659]) \\ or

gp: E = ellinit("286650ba2")

$$y^2 + x y = x^{3} - x^{2} - 78500067 x - 226972151659$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-\frac{431581}{121}, \frac{118672133}{1331}\right)$$ $$\hat{h}(P)$$ ≈ 5.6878702772

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(10054, -5027\right)$$, $$\left(-3386, 1693\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-3386, 1693\right)$$, $$\left(10054, -5027\right)$$, $$\left(11029, 493198\right)$$, $$\left(11029, -504227\right)$$, $$\left(240550, 117778429\right)$$, $$\left(240550, -118018979\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$286650$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$8700308288012250000000000$$ = $$2^{10} \cdot 3^{6} \cdot 5^{12} \cdot 7^{10} \cdot 13^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{39920686684059609}{6492304000000}$$ = $$2^{-10} \cdot 3^{3} \cdot 5^{-6} \cdot 7^{-4} \cdot 13^{-2} \cdot 37^{3} \cdot 3079^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$5.6878702772$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.0512789540124$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$256$$  = $$2\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 286650.2.a.ba

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - q^{2} + q^{4} - q^{8} - 4q^{11} - q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 70778880 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$4.66668861397$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{10}$$ Non-split multiplicative 1 1 10 10
$$3$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$5$$ $$4$$ $$I_6^{*}$$ Additive 1 2 12 6
$$7$$ $$4$$ $$I_4^{*}$$ Additive -1 2 10 4
$$13$$ $$2$$ $$I_{2}$$ 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

## $p$-adic data

### $p$-adic regulators

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

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 286650.ba consists of 4 curves linked by isogenies of degrees dividing 4.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-21}, \sqrt{65})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{26}, \sqrt{210})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-10}, \sqrt{21})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.