Properties

Label 2.0.3.1-54684.5-d1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 54684 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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(Polrev([1, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-312a+219\right){x}-476a-759\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([1,1]),K([219,-312]),K([-759,-476])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([1,1]),Polrev([219,-312]),Polrev([-759,-476])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![1,1],K![219,-312],K![-759,-476]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-270a+138)\) = \((-2a+1)^{2}\cdot(2)\cdot(3a-2)^{2}\cdot(-6a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 54684 \) = \(3^{2}\cdot4\cdot7^{2}\cdot31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1719435168a+1404692928)\) = \((-2a+1)^{6}\cdot(2)^{5}\cdot(3a-2)^{6}\cdot(-6a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2514341098284309504 \) = \(3^{6}\cdot4^{5}\cdot7^{6}\cdot31^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{511363962461}{916132832} a + \frac{1018073036305}{916132832} \)
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 $\left(-a + 16 : -52 a + 89 : 1\right)$
Height \(0.084171321110073382198084056813547697187\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.084171321110073382198084056813547697187 \)
Period: \( 0.40199203524717536373138590208280111105 \)
Tamagawa product: \( 50 \)  =  \(1\cdot5\cdot2\cdot5\)
Torsion order: \(1\)
Leading coefficient: \( 3.9070679144770288157252849241831210195 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((2)\) \(4\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((3a-2)\) \(7\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((-6a+1)\) \(31\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5Cs.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 54684.5-d consists of curves linked by isogenies of degrees dividing 25.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.