Properties

Label 2.0.3.1-20956.2-b1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((164a-54)\)
Conductor norm \( 20956 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-857a-14246\right){x}-66136a-659109\)
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([0,1]),K([-14246,-857]),K([-659109,-66136])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-14246,-857])),Pol(Vecrev([-659109,-66136]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,1],K![0,1],K![-14246,-857],K![-659109,-66136]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((164a-54)\) = \((2)\cdot(-4a+1)^{2}\cdot(6a-5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 20956 \) = \(4\cdot13^{2}\cdot31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-456575156224a+118178709504)\) = \((2)^{25}\cdot(-4a+1)^{6}\cdot(6a-5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 168469617906861312311296 \) = \(4^{25}\cdot13^{6}\cdot31\)
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: \(1\)
Generator $\left(-13 a - 98 : 614 a - 415 : 1\right)$
Height \(0.278764670782865\)
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: \( 0.278764670782865 \)
Period: \( 0.102184592010966 \)
Tamagawa product: \( 50 \)  =  \(5^{2}\cdot2\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 3.28921692441585 \)
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\((j)_{-}\)
\((2)\) \(4\) \(25\) \(I_{25}\) Split multiplicative \(-1\) \(1\) \(25\) \(25\)
\((-4a+1)\) \(13\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((6a-5)\) \(31\) \(1\) \(I_{1}\) 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
\(5\) 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 20956.2-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.