Properties

Label 2.0.3.1-54684.6-b1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-258a+60)\)
Conductor norm \( 54684 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22958a+1487\right){x}-108899a+1456079\)
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([1,1]),K([1487,22958]),K([1456079,-108899])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([1,1])),Pol(Vecrev([1,1])),Pol(Vecrev([1487,22958])),Pol(Vecrev([1456079,-108899]))], K);
 
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,1],K![1487,22958],K![1456079,-108899]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-258a+60)\) = \((-2a+1)^{2}\cdot(2)\cdot(3a-2)^{2}\cdot(6a-5)\)
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: \((-1664266272768a-125023813632)\) = \((-2a+1)^{6}\cdot(2)^{25}\cdot(3a-2)^{6}\cdot(6a-5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2993486096968749756186624 \) = \(3^{6}\cdot4^{25}\cdot7^{6}\cdot31\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.0803984070494351 \)
Tamagawa product: \( 25 \)  =  \(1\cdot5^{2}\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.32090209762042 \)
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 \(-1\) \(2\) \(6\) \(0\)
\((2)\) \(4\) \(25\) \(I_{25}\) Split multiplicative \(-1\) \(1\) \(25\) \(25\)
\((3a-2)\) \(7\) \(1\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((6a-5)\) \(31\) \(1\) \(I_{1}\) Non-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
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 54684.6-b consists of curves linked by isogenies of degrees dividing 25.

Base change

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