Label 64.a3
Conductor $64$
Discriminant $4096$
j-invariant \( 1728 \)
CM yes (\(D=-4\))
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -4, 0])
gp: E = ellinit([0, 0, 0, -4, 0])
magma: E := EllipticCurve([0, 0, 0, -4, 0]);

\(y^2=x^3-4x\)  Toggle raw display

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{2}\Z\)

Torsion generators

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

\( \left(0, 0\right) \), \( \left(2, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-2, 0\right) \), \( \left(0, 0\right) \), \( \left(2, 0\right) \)  Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 64 \)  =  \(2^{6}\)
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: \(4096 \)  =  \(2^{12} \)
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: $N(\mathrm{U}(1))$
Faltings height: \(-0.61738574535156420883504296185\dots\)
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()
magma: RealPeriod(E);
Real period: \(3.7081493546027438368677006944\dots\)
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: \( 4 \)  = \( 2^{2} \)
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

Modular form   64.2.a.a

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);

\( q + 2q^{5} - 3q^{9} - 6q^{13} + 2q^{17} + O(q^{20}) \)  Toggle raw display

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

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

\( L(E,1) \) ≈ \( 0.92703733865068595921692517359767012944 \)

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\((j)_{-}\)
\(2\) \(4\) \(I_2^{*}\) Additive -1 6 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 \( p < 1000 \) 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(5,20) if E.conductor().valuation(p)<2]

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2
Reduction type add
$\lambda$-invariant(s) -
$\mu$-invariant(s) -

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.


This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 64.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/4\Z\)
$4$ \(\Q(\zeta_{8})\) \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.0.4194304.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$8$ 8.4.67108864.1 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.2.573308928.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$8$ 8.0.32768000.1 \(\Z/2\Z \times \Z/10\Z\) Not in database
$16$ 16.0.18014398509481984.1 \(\Z/8\Z \times \Z/8\Z\) Not in database
$16$ 16.0.328683126924509184.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ 16.4.16777216000000000000.2 \(\Z/2\Z \times \Z/10\Z\) Not in database
$16$ 16.0.17179869184000000.1 \(\Z/4\Z \times \Z/20\Z\) Not in database
$16$ 16.4.5258930030792146944.1 \(\Z/2\Z \times \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive.