Properties

Label 2.0.3.1-96957.4-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-324a+27)\)
Conductor norm \( 96957 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

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

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(40a-55\right){x}-102a+122\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-55,40]),K([122,-102])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([-55,40])),Pol(Vecrev([122,-102]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-55,40],K![122,-102]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-324a+27)\) = \((-2a+1)^{6}\cdot(3a-2)\cdot(-5a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 96957 \) = \(3^{6}\cdot7\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((336663a-3414069)\) = \((-2a+1)^{6}\cdot(3a-2)^{9}\cdot(-5a+2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10619818400583 \) = \(3^{6}\cdot7^{9}\cdot19^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{53878632609825}{14567652127} a + \frac{12300435956106}{14567652127} \)
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: \(0\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2 a + 2 : -a - 10 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.07647083172767 \)
Tamagawa product: \( 18 \)  =  \(1\cdot3^{2}\cdot2\)
Torsion order: \(3\)
Leading coefficient: \( 2.48600289789100 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(II\) Additive \(-1\) \(6\) \(6\) \(0\)
\((3a-2)\) \(7\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((-5a+2)\) \(19\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 96957.4-a consists of curves linked by isogenies of degree 3.

Base change

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