Properties

Label 6.6.371293.1-79.3-d8
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{13})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(85a^{5}-76a^{4}-434a^{3}+346a^{2}+492a-365\right){x}-795a^{5}+975a^{4}+3695a^{3}-4054a^{2}-3484a+3086\)
sage: E = EllipticCurve([K([-2,-3,1,1,0,0]),K([1,1,-3,0,1,0]),K([2,-3,-4,1,1,0]),K([-365,492,346,-434,-76,85]),K([3086,-3484,-4054,3695,975,-795])])
 
gp: E = ellinit([Polrev([-2,-3,1,1,0,0]),Polrev([1,1,-3,0,1,0]),Polrev([2,-3,-4,1,1,0]),Polrev([-365,492,346,-434,-76,85]),Polrev([3086,-3484,-4054,3695,975,-795])], K);
 
magma: E := EllipticCurve([K![-2,-3,1,1,0,0],K![1,1,-3,0,1,0],K![2,-3,-4,1,1,0],K![-365,492,346,-434,-76,85],K![3086,-3484,-4054,3695,975,-795]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-4a^3+3a-2)\) = \((a^5-4a^3+3a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 79 \) = \(79\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^5-a^4+21a^3+2a^2-27a-1)\) = \((a^5-4a^3+3a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -493039 \) = \(-79^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3428757028066412689128269890262938}{493039} a^{5} - \frac{997049030798831835894322909079689}{493039} a^{4} - \frac{17850902058216642220750712826565565}{493039} a^{3} + \frac{1054993896343092497695425618836686}{493039} a^{2} + \frac{21320754151285089843137628848091614}{493039} a + \frac{4834616150757005557838566061341763}{493039} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{19}{4} a^{5} + 5 a^{4} + \frac{39}{2} a^{3} - \frac{67}{4} a^{2} - \frac{59}{4} a + \frac{39}{4} : -\frac{17}{8} a^{5} + 11 a^{3} - \frac{9}{8} a^{2} - \frac{97}{8} a + \frac{7}{2} : 1\right)$
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: \( 202.53939857027286036588199073950547562 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 1.32957 \)
Analytic order of Ш: \( 16 \) (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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 79.3-d consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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