# Properties

 Label 32a2 Conductor $32$ Discriminant $64$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x$$ y^2=x^3-x (homogenize, simplify) $$y^2z=x^3-xz^2$$ y^2z=x^3-xz^2 (dehomogenize, simplify) $$y^2=x^3-x$$ y^2=x^3-x (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -1, 0])

gp: E = ellinit([0, 0, 0, -1, 0])

magma: E := EllipticCurve([0, 0, 0, -1, 0]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(0, 0\right)$$, $$\left(1, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1, 0\right)$$, $$\left(0, 0\right)$$, $$\left(1, 0\right)$$

## 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: $64$ = $2^{6}$ 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: $N(\mathrm{U}(1))$ Faltings height: $-0.96395933563153686354365902258\dots$ Stable Faltings height: $-1.3105329259115095182522750833\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $5.2441151085842396209296791798\dots$ 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: $2$  = $2$ 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) 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); Special value: $L(E,1)$ ≈ $0.65551438857302995261620989747$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2 $\Gamma_0(N)$-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)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $III$ Additive -1 5 6 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
$2$ 2Cs 16.384.9.568

## $p$-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.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.0.4.1-64.1-CMa1 $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.2.8.1-32.1-a4 $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.4194304.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.67108864.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.143327232.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.8192000.1 $$\Z/2\Z \oplus \Z/20\Z$$ Not in database $16$ 16.0.18014398509481984.1 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.20542695432781824.1 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $16$ 16.4.1048576000000000000.1 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ 16.4.5258930030792146944.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ 16.0.17179869184000000.1 $$\Z/4\Z \oplus \Z/20\Z$$ Not in database

We only show fields where the torsion growth is primitive.