Properties

Label 6.6.371293.1-79.3-a2
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){x}{y}+\left(a^{4}-4a^{2}+a+3\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+a+1\right){x}^{2}+\left(2a^{5}-11a^{4}+26a^{2}+4a-7\right){x}-5a^{5}+16a^{4}-12a^{3}-22a^{2}+23a\)
sage: E = EllipticCurve([K([3,-2,-4,1,1,0]),K([1,1,-4,-4,1,1]),K([3,1,-4,0,1,0]),K([-7,4,26,0,-11,2]),K([0,23,-22,-12,16,-5])])
 
gp: E = ellinit([Polrev([3,-2,-4,1,1,0]),Polrev([1,1,-4,-4,1,1]),Polrev([3,1,-4,0,1,0]),Polrev([-7,4,26,0,-11,2]),Polrev([0,23,-22,-12,16,-5])], K);
 
magma: E := EllipticCurve([K![3,-2,-4,1,1,0],K![1,1,-4,-4,1,1],K![3,1,-4,0,1,0],K![-7,4,26,0,-11,2],K![0,23,-22,-12,16,-5]]);
 

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: \((-4a^4+a^3+17a^2-2a-9)\) = \((a^5-4a^3+3a-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -6241 \) = \(-79^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{308721461939500395175}{6241} a^{5} + \frac{855440019384751139506}{6241} a^{4} + \frac{28698270198869399627}{6241} a^{3} - \frac{1285707960303197449691}{6241} a^{2} + \frac{424546949726100418011}{6241} a + \frac{174329076201954485036}{6241} \)
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 $\left(a^{5} - 4 a^{4} - 3 a^{3} + 14 a^{2} - 2 a - 3 : 6 a^{5} - 10 a^{4} - 16 a^{3} + 33 a^{2} - 9 a - 1 : 1\right)$
Height \(0.038527218061788762413655668891111412806\)
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: $\left(-a^{5} - \frac{1}{4} a^{4} + \frac{13}{4} a^{3} + 2 a^{2} - \frac{3}{2} a - \frac{3}{2} : \frac{1}{4} a^{5} + a^{4} - \frac{1}{2} a^{3} - \frac{7}{4} a^{2} - \frac{3}{4} a + \frac{3}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.038527218061788762413655668891111412806 \)
Period: \( 13265.049995389334293006990187993138672 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 2.51617 \)
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\((j)_{-}\)
\((a^5-4a^3+3a-2)\) \(79\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 79.3-a consists of curves linked by isogenies of degree 2.

Base change

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

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