# Properties

 Label 32.a3 Conductor $32$ Discriminant $64$ j-invariant $1728$ CM yes ($D=-4$) Rank $0$ Torsion Structure $\Z/{2}\Z \times \Z/{2}\Z$

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Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -1, 0]); // or
magma: E := EllipticCurve("32a2");
sage: E = EllipticCurve([0, 0, 0, -1, 0]) # or
sage: E = EllipticCurve("32a2")
gp: E = ellinit([0, 0, 0, -1, 0]) \\ or
gp: E = ellinit("32a2")

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

## Mordell-Weil group structure

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

## Torsion generators

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

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

## Integral points

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

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

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $32$ = $2^{5}$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $64$ = $2^{6}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $1728$ = $2^{6} \cdot 3^{3}$ $\text{End} (E)$ = $\Z[\sqrt{-1}]$ (Complex Multiplication) $\text{ST} (E)$ = $N(\mathrm{U}(1))$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $5.24411510858$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $2$  = $2$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $4$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form32.2.1.a

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

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

For more coefficients, see the Downloads section to the right.

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2 : curve is not $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

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$(j)_{-}$
$2$ $2$ $III$ Additive -1 5 6 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 $p$ except those listed.

prime Image of Galois representation
$2$ Cs

For all other primes $p$, the image is 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 32.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{2})$ $\Z/2\Z \times \Z/4\Z$ 2.2.8.1-32.1-a4
$\Q(\sqrt{-1})$ $\Z/2\Z \times \Z/4\Z$ 2.0.4.1-64.1-CMa1
4 $\Q(\zeta_{8})$ $\Z/4\Z \times \Z/4\Z$ Not in database

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.