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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -1335707, -463815699]) # or

sage: E = EllipticCurve("335730v4")

gp: E = ellinit([1, 1, 0, -1335707, -463815699]) \\ or

gp: E = ellinit("335730v4")

magma: E := EllipticCurve([1, 1, 0, -1335707, -463815699]); // or

magma: E := EllipticCurve("335730v4");

$$y^2 + x y = x^{3} + x^{2} - 1335707 x - 463815699$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(35642, -6743251\right)$$ $$\left(\frac{1009177}{441}, \frac{841702079}{9261}\right)$$ $$\hat{h}(P)$$ ≈ $3.2560553010321374$ $7.906733511217537$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{3637}{4}, \frac{3637}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-878, 6189\right)$$, $$\left(-878, -5311\right)$$, $$\left(-758, 11049\right)$$, $$\left(-758, -10291\right)$$, $$\left(-458, 7449\right)$$, $$\left(-458, -6991\right)$$, $$\left(-403, 3239\right)$$, $$\left(-403, -2836\right)$$, $$\left(1347, 12864\right)$$, $$\left(1347, -14211\right)$$, $$\left(1622, 39689\right)$$, $$\left(1622, -41311\right)$$, $$\left(35642, 6707609\right)$$, $$\left(35642, -6743251\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$335730$$ = $$2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 31$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$59804429890443750000$$ = $$2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 19^{6} \cdot 31$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5601911201812801}{1271193750000}$$ = $$2^{-4} \cdot 3^{-8} \cdot 5^{-8} \cdot 31^{-1} \cdot 177601^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.89310034155$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.142719556225$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$64$$  = $$2\cdot2\cdot2^{3}\cdot2\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 335730.2.a.v

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 6q^{13} - q^{15} + q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 14155776 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

sage: r = E.rank();

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

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

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

$$L^{(2)}(E,1)/2!$$ ≈ $$8.88994484936$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$I_{8}$$ Non-split multiplicative 1 1 8 8
$$5$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$19$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$31$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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 X102.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 24.

sage: rho = E.galois_representation();

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

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

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

prime Image of Galois representation
$$2$$ B

## $p$-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, 4 and 8.
Its isogeny class 335730.v consists of 6 curves linked by isogenies of degrees dividing 8.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ $$\Q(\sqrt{-19})$$ $$\Z/8\Z$$ Not in database
$2$ $$\Q(\sqrt{31})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$2$ $$\Q(\sqrt{-589})$$ $$\Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{-19}, \sqrt{31})$$ $$\Z/2\Z \times \Z/8\Z$$ 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.