Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -337, 2272]); // or

magma: E := EllipticCurve("2541j2");

sage: E = EllipticCurve([0, 1, 1, -337, 2272]) # or

sage: E = EllipticCurve("2541j2")

gp: E = ellinit([0, 1, 1, -337, 2272]) \\ or

gp: E = ellinit("2541j2")

$$y^2 + y = x^{3} + x^{2} - 337 x + 2272$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(10, 1\right)$$ $$\hat{h}(P)$$ ≈ 0.870906460922

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(10, 1\right)$$, $$\left(10, -2\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$2541$$ = $$3 \cdot 7 \cdot 11^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$124509$$ = $$3 \cdot 7^{3} \cdot 11^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{35084566528}{1029}$$ = $$2^{18} \cdot 3^{-1} \cdot 7^{-3} \cdot 11 \cdot 23^{3}$$ 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: $$0.870906460922$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$3.07518673438$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1$$  = $$1\cdot1\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

#### Modular form2541.2.a.e

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 576 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$2.67819999551$$

## 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$$(j)_{-}$$
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$11$$ $$1$$ $$II$$ Additive -1 2 2 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
$$3$$ 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.

## 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 split ordinary nonsplit add ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4,1 2 1 1 - 1 1 1 1,1 1 1 1 1 1 1 0,0 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 2541.e consists of 2 curves linked by isogenies of degree 3.

## 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
2 $$\Q(\sqrt{33})$$ $$\Z/3\Z$$ Not in database
3 3.3.10164.1 $$\Z/2\Z$$ Not in database
6 6.6.3409127568.1 $$\Z/6\Z$$ Not in database
6.6.2169444816.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.0.9509900499.1 $$\Z/3\Z$$ 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.