Properties

Label 2.2.12.1-1024.1-k4
Base field \(\Q(\sqrt{3}) \)
Conductor \((32)\)
Conductor norm \( 1024 \)
CM yes (\(-48\))
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

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

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

Weierstrass equation

\(y^2=x^{3}+ax^{2}+\left(170a-299\right)x-1711a+2958\)
sage: E = EllipticCurve(K, [0, a, 0, 170*a - 299, -1711*a + 2958])
 
gp: E = ellinit([0, a, 0, 170*a - 299, -1711*a + 2958],K)
 
magma: E := ChangeRing(EllipticCurve([0, a, 0, 170*a - 299, -1711*a + 2958]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((32)\) = \( \left(a + 1\right)^{10} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 1024 \) = \( 2^{10} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((8192)\) = \( \left(a + 1\right)^{26} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 67108864 \) = \( 2^{26} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -818626500 a + 1417905000 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-12}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-\frac{14}{3} a + \frac{19}{3} : -\frac{8}{9} a : 1\right)$
Height \(0.429957487887703\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-5 a + 6 : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.429957487887703 \)
Period: \( 10.8359497894950 \)
Tamagawa product: \( 4 \)
Torsion order: \(2\)
Leading coefficient: \(2.68987360532914\)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(4\) \(I_{12}^*\) Additive \(-1\) \(10\) \(26\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

The image is a Borel subgroup if \(p\in \{ 2, 3\}\), 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, 4, 6 and 12.
Its isogeny class 1024.1-k 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.