Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -11, -14])

gp: E = ellinit([0, 0, 0, -11, -14])

magma: E := EllipticCurve([0, 0, 0, -11, -14]);

$$y^2=x^3-11x-14$$ ## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-2, 0\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2, 0\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$32$$ = $2^{5}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $512$ = $2^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$287496$$ = $2^{3} \cdot 3^{3} \cdot 11^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-4}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.61738574535156420883504296185\dots$ Stable Faltings height: $-1.1372461307715231908979670529\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $2.6220575542921198104648395899\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1$ 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: $1$ (exact) 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); Special value: $L(E,1)$ ≈ $0.65551438857302995261620989747275764966$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4 $\Gamma_0(N)$-optimal: no Manin constant: 2

## Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

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$(j)_{-}$
$2$ $1$ $I_0^{*}$ Additive -1 5 9 0

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 16.192.3.554

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ 2 add - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 and 4.
Its isogeny class 32.a consists of 3 curves linked by isogenies of degrees dividing 4.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.2.8.1-32.1-a5 $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-64.1-CMa2 $2$ $$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ 2.0.8.1-32.1-a3 $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.2048.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.0.512.1 $$\Z/8\Z$$ Not in database $8$ 8.0.16777216.2 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ $$\Q(\zeta_{16})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.4194304.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.143327232.1 $$\Z/6\Z$$ Not in database $8$ 8.0.8192000.1 $$\Z/20\Z$$ Not in database $16$ 16.4.73786976294838206464.5 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ 16.0.18014398509481984.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ 16.0.288230376151711744.2 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ 16.0.20542695432781824.1 $$\Z/3\Z \times \Z/12\Z$$ Not in database $16$ 16.4.1048576000000000000.1 $$\Z/10\Z$$ Not in database $16$ 16.4.5258930030792146944.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.17179869184000000.1 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ 16.0.5258930030792146944.3 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/40\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.