Label 64.a4
Conductor $64$
Discriminant $-64$
j-invariant \( 1728 \)
CM yes (\(D=-4\))
Rank $0$
Torsion structure \(\Z/{2}\Z\)

Related objects


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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 1, 0])
gp: E = ellinit([0, 0, 0, 1, 0])
magma: E := EllipticCurve([0, 0, 0, 1, 0]);

\(y^2=x^3+x\)  Toggle raw display

Mordell-Weil group structure


Torsion generators

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

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

Integral points

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

\( \left(0, 0\right) \)  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: $-64 $  =  $-1 \cdot 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()
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: $ 1 $
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
Torsion order: $2$
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) $ ≈ $ 0.92703733865068595921692517359767012944 $

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: 4
$ \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$(j)_{-}$
$2$ $1$ $II$ Additive -1 6 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
$2$ 2B 16.384.9.600

$p$-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 and 4.
Its isogeny class 64.a consists of 3 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-1}) \) \(\Z/2\Z \times \Z/2\Z\)
$2$ \(\Q(\sqrt{2}) \) \(\Z/4\Z\)
$2$ \(\Q(\sqrt{-2}) \) \(\Z/4\Z\)
$4$ \(\Q(\zeta_{8})\) \(\Z/4\Z \times \Z/4\Z\) 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
$8$ 8.2.573308928.1 \(\Z/6\Z\) 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
$16$ 16.4.4611686018427387904.2 \(\Z/16\Z\) Not in database
$16$ 16.0.328683126924509184.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ 16.4.16777216000000000000.2 \(\Z/10\Z\) Not in database
$16$ 16.4.5258930030792146944.1 \(\Z/12\Z\) Not in database
$16$ 16.0.17179869184000000.1 \(\Z/4\Z \times \Z/20\Z\) Not in database
$16$ 16.0.5258930030792146944.3 \(\Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive.