Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -456, 9115]); // or

magma: E := EllipticCurve("35739g1");

sage: E = EllipticCurve([0, 0, 1, -456, 9115]) # or

sage: E = EllipticCurve("35739g1")

gp: E = ellinit([0, 0, 1, -456, 9115]) \\ or

gp: E = ellinit("35739g1")

$$y^2 + y = x^{3} - 456 x + 9115$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-19, 104\right)$$ $$\left(\frac{1}{4}, \frac{755}{8}\right)$$ $$\hat{h}(P)$$ ≈ 0.295248009628 2.40166446161

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-19, 104\right)$$, $$\left(19, 85\right)$$, $$\left(25, 115\right)$$, $$\left(47, 302\right)$$, $$\left(289, 4900\right)$$, $$\left(148611521, 1811668439300\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$35739$$ = $$3^{2} \cdot 11 \cdot 19^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-29825517843$$ = $$-1 \cdot 3^{3} \cdot 11^{5} \cdot 19^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{56623104}{161051}$$ = $$-1 \cdot 2^{21} \cdot 3^{3} \cdot 11^{-5}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.341518756178$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$1.03673183328$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$20$$  = $$2\cdot5\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 35739.2.a.i

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 24000 $$\Gamma_0(N)$$-optimal: yes 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)

$$L^{(2)}(E,1)/2!$$ ≈ $$7.08126732382$$

## 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$$(j)_{-}$$
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$11$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$19$$ $$2$$ $$III$$ Additive 1 2 3 0

## 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
$$5$$ Ns

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 ss add ordinary ordinary split ordinary ordinary add ordinary ordinary ordinary ss ordinary ordinary ordinary 2,5 - 4 2 3 2 2 - 2 2 2 2,2 2 2 2 0,0 - 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 no rational isogenies. Its isogeny class 35739.i consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2508.1 $$\Z/2\Z$$ Not in database
6 6.0.3943870128.1 $$\Z/2\Z \times \Z/2\Z$$ 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.