# Properties

 Label 32a1 Conductor $32$ Discriminant $-4096$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

This is a model for the modular curve $X_0(32)$.

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 4, 0]) # or

sage: E = EllipticCurve("32.a4")

gp: E = ellinit([0, 0, 0, 4, 0]) \\ or

gp: E = ellinit("32.a4")

magma: E := EllipticCurve([0, 0, 0, 4, 0]); // or

magma: E := EllipticCurve("32.a4");

$$y^2=x^3+4x$$

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 4\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(0, 0\right)$$, $$(2,\pm 4)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$32$$ = $$2^{5}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-4096$$ = $$-1 \cdot 2^{12}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$1728$$ = $$2^{6} \cdot 3^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$

## 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[1]  magma: RealPeriod(E); Real period: $$2.6220575542921198104648395899$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$4$$  = $$2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+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: 1 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L(E,1)$$ ≈ $$0.65551438857302995261620989747275764966$$

## Local data

This elliptic curve is semistable. There is only one prime of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_3^{*}$$ Additive -1 5 12 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 mod $$p$$ Galois representation has maximal image for all primes $$p < 1000$$ .

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 less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.0.4.1-64.1-CMa1 $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database $4$ 4.2.1024.1 $$\Z/8\Z$$ Not in database $8$ 8.0.4194304.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.2.143327232.1 $$\Z/12\Z$$ Not in database $8$ 8.0.8192000.1 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ 16.0.18014398509481984.1 $$\Z/8\Z \times \Z/8\Z$$ Not in database $16$ 16.4.4611686018427387904.2 $$\Z/16\Z$$ Not in database $16$ 16.0.20542695432781824.1 $$\Z/6\Z \times \Z/12\Z$$ Not in database $16$ 16.4.1048576000000000000.1 $$\Z/20\Z$$ Not in database $16$ 16.0.17179869184000000.1 $$\Z/4\Z \times \Z/20\Z$$ Not in database

We only show fields where the torsion growth is primitive.