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Properties

Label 32.a4
Conductor \(32\)
Discriminant \(-4096\)
j-invariant \( 1728 \)
CM yes (\(D=-4\))
Rank \(0\)
Torsion Structure \(\Z/{4}\Z\)

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

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, 4, 0]); // or
magma: E := EllipticCurve("32a1");
sage: E = EllipticCurve([0, 0, 0, 4, 0]) # or
sage: E = EllipticCurve("32a1")
gp: E = ellinit([0, 0, 0, 4, 0]) \\ or
gp: E = ellinit("32a1")

\( y^2 = x^{3} + 4 x \)

Mordell-Weil group structure

\(\Z/{4}\Z\)

Torsion generators

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

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

Integral points

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

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

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
\(\Delta\)  =  \(-4096 \)  =  \(-1 \cdot 2^{12} \)
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]
\( \Omega \)  ≈  \(2.62205755429\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 4 \)  = \( 2^{2} \)
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 form 32.2.1.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()
1 : curve is \( \Gamma_0(N) \)-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\((j)_{-}\)
\(2\) \(4\) \( I_3^{*} \) Additive -1 5 12 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 \( p \) .

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
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.

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.