Properties

Label 2.0.3.1-130816.2-a2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((400a-96)\)
Conductor norm \( 130816 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-a{x}^{2}+\left(2a+21\right){x}+47a-54\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([21,2]),K([-54,47])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([21,2])),Pol(Vecrev([-54,47]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![21,2],K![-54,47]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((400a-96)\) = \((2)^{4}\cdot(-3a+1)\cdot(9a-8)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 130816 \) = \(4^{4}\cdot7\cdot73\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1130496a-606208)\) = \((2)^{14}\cdot(-3a+1)^{2}\cdot(9a-8)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 960193626112 \) = \(4^{14}\cdot7^{2}\cdot73\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{20198991}{14308} a - \frac{3200392}{3577} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-2 a + 1 : -6 a + 2 : 1\right)$
Height \(0.499940384488220\)
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: $\left(-4 a + 1 : 0 : 1\right)$
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: \( 0.499940384488220 \)
Period: \( 1.36258781394617 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.14638657307919 \)
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\((j)_{-}\)
\((2)\) \(4\) \(4\) \(I_{6}^{*}\) Additive \(-1\) \(4\) \(14\) \(2\)
\((-3a+1)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((9a-8)\) \(73\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 130816.2-a consists of curves linked by isogenies of degree 2.

Base change

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