# Properties

 Label 32a3 Conductor 32 Discriminant 512 j-invariant $$287496$$ CM yes ($$D=-16$$) Rank 0 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -11, -14]); // or
magma: E := EllipticCurve("32a3");
sage: E = EllipticCurve([0, 0, 0, -11, -14]) # or
sage: E = EllipticCurve("32a3")
gp: E = ellinit([0, 0, 0, -11, -14]) \\ or
gp: E = ellinit("32a3")

$$y^2 = x^{3} - 11 x - 14$$

## Mordell-Weil group structure

$$\Z/{2}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

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

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

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

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$32$$ = $$2^{5}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$512$$ = $$2^{9}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$287496$$ = $$2^{3} \cdot 3^{3} \cdot 11^{3}$$ Endomorphism ring: $$\Z[\sqrt{-4}]$$ ( Complex Multiplication) Sato-Tate Group: $N(\mathrm{U}(1))$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.62205755429$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$1$$  = $$1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form32.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 4 $$\Gamma_0(N)$$-optimal: no Manin constant: not computed

#### 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[2]/factorial(ar[1])

$$L(E,1)$$ ≈ $$0.655514388573$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
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

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 for all primes $$p$$ .

The image is a Borel subgroup if $$p=2$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=-1$$.

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 32a consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.2.8.1-32.1-a5
$$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-64.1-CMa2
$$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ 2.0.8.1-32.1-a3
4 $$\Q(\zeta_{8})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
4.2.2048.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database
4.0.512.1 $$\Z/8\Z$$ 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.