Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-1024.1-j6
Conductor \((32)\)
Conductor norm \( 1024 \)
CM yes (\(-12\))
base-change no
Q-curve yes
Torsion order \( 4 \)
Rank not available

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Base field \(\Q(\sqrt{3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)
gp (2.8): K = nfinit(a^2 - 3);

Weierstrass equation

\( y^2 = x^{3} + a x^{2} + \left(-10 a - 19\right) x - 31 a - 54 \)
magma: E := ChangeRing(EllipticCurve([0, a, 0, -10*a - 19, -31*a - 54]),K);
sage: E = EllipticCurve(K, [0, a, 0, -10*a - 19, -31*a - 54])
gp (2.8): E = ellinit([0, a, 0, -10*a - 19, -31*a - 54],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((32)\) = \( \left(a + 1\right)^{10} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 1024 \) = \( 2^{10} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((2048)\) = \( \left(a + 1\right)^{22} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 4194304 \) = \( 2^{22} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 54000 \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z[\sqrt{-3}]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $N(\mathrm{U}(1))$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
Generators: $\left(-2 a - 1 : 0 : 1\right)$,$\left(-a - 2 : 0 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(4\) \(I_{8}^*\) Additive \(-1\) \(10\) \(22\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 1024.1-j consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.