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## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -81291, -2802310]); // or
magma: E := EllipticCurve("77616gn2");
sage: E = EllipticCurve([0, 0, 0, -81291, -2802310]) # or
sage: E = EllipticCurve("77616gn2")
gp: E = ellinit([0, 0, 0, -81291, -2802310]) \\ or
gp: E = ellinit("77616gn2")

$$y^2 = x^{3} - 81291 x - 2802310$$

## 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()

 $$P$$ = $$\left(-185, 2430\right)$$ $$\hat{h}(P)$$ ≈ 2.37108974523

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(-35, 0\right)$$, $$\left(301, 0\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-266, 0\right)$$, $$\left(-185, 2430\right)$$, $$\left(-91, 1960\right)$$, $$\left(-35, 0\right)$$, $$\left(301, 0\right)$$, $$\left(1351, 48510\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$77616$$ = $$2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$30987648070815744$$ = $$2^{12} \cdot 3^{12} \cdot 7^{6} \cdot 11^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{169112377}{88209}$$ = $$3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}$$ 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: $$2.37108974523$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.299464050369$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$128$$  = $$2^{2}\cdot2^{2}\cdot2^{2}\cdot2$$ 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 77616.2.a.bz

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - 2q^{5} + q^{11} + 2q^{13} - 2q^{17} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 442368 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$5.68044911116$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_4^{*}$$ Additive -1 4 12 0
$$3$$ $$4$$ $$I_6^{*}$$ Additive -1 2 12 6
$$7$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$11$$ $$2$$ $$I_{2}$$ 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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ordinary add split ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss ordinary - - 1 - 4 1 1 1,1 1 1 3 1 1 1,1 1 - - 0 - 0 0 0 0,0 0 0 0 0 0 0,0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 77616.bz 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 $$\Q(\sqrt{-7}, \sqrt{33})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-11}, \sqrt{-21})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{7})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.