Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -55, 98]); // or

magma: E := EllipticCurve("11913e1");

sage: E = EllipticCurve([1, 1, 1, -55, 98]) # or

sage: E = EllipticCurve("11913e1")

gp: E = ellinit([1, 1, 1, -55, 98]) \\ or

gp: E = ellinit("11913e1")

$$y^2 + x y + y = x^{3} + x^{2} - 55 x + 98$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-4, 18\right)$$ $$\hat{h}(P)$$ ≈ 0.139799000295

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-4, 18\right)$$, $$\left(-4, -15\right)$$, $$\left(2, 0\right)$$, $$\left(2, -3\right)$$, $$\left(7, 7\right)$$, $$\left(7, -15\right)$$, $$\left(151, 1785\right)$$, $$\left(151, -1937\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$11913$$ = $$3 \cdot 11 \cdot 19^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$4324419$$ = $$3^{2} \cdot 11^{3} \cdot 19^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{51026761}{11979}$$ = $$3^{-2} \cdot 11^{-3} \cdot 19 \cdot 139^{3}$$ 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: $$0.139799000295$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$2.31180899009$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$6$$  = $$2\cdot3\cdot1$$ 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 form 11913.2.a.c

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 3q^{7} + 3q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 6q^{13} + 3q^{14} - q^{15} - q^{16} - q^{18} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 2160 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$1.93913151413$$

## 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$$(j)_{-}$$
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$19$$ $$1$$ $$II$$ 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 $$p$$ .

## $p$-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 ordinary nonsplit ordinary ordinary split ordinary ss add ordinary ordinary ordinary ordinary ordinary ordinary ordinary 2 1 1 1 2 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

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 11913.c 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 3.3.15884.1 $$\Z/2\Z$$ Not in database
6 6.6.11101264064.1 $$\Z/2\Z \times \Z/2\Z$$ 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.