Properties

Label 6.6.371293.1-79.4-c1
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 10 \)
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

\({y}^2+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){x}^{2}+\left(-6a^{5}+13a^{4}+16a^{3}-44a^{2}+9a+11\right){x}+15a^{5}-33a^{4}-38a^{3}+107a^{2}-30a-14\)
sage: E = EllipticCurve([K([3,2,-4,-4,1,1]),K([2,-2,-4,1,1,0]),K([-2,1,1,0,0,0]),K([11,9,-44,16,13,-6]),K([-14,-30,107,-38,-33,15])])
 
gp: E = ellinit([Polrev([3,2,-4,-4,1,1]),Polrev([2,-2,-4,1,1,0]),Polrev([-2,1,1,0,0,0]),Polrev([11,9,-44,16,13,-6]),Polrev([-14,-30,107,-38,-33,15])], K);
 
magma: E := EllipticCurve([K![3,2,-4,-4,1,1],K![2,-2,-4,1,1,0],K![-2,1,1,0,0,0],K![11,9,-44,16,13,-6],K![-14,-30,107,-38,-33,15]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+3a^2-3a-2)\) = \((-a^4+a^3+3a^2-3a-2)\)
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: \((2a^5+a^4-12a^3-4a^2+15a+1)\) = \((-a^4+a^3+3a^2-3a-2)^{2}\)
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: \( -\frac{506078693721616}{6241} a^{5} - \frac{251520007112219}{6241} a^{4} + \frac{27264105978348}{79} a^{3} + \frac{1200019893455926}{6241} a^{2} - \frac{1240009899654337}{6241} a - \frac{338045087886387}{6241} \)
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: \(0\)
Torsion structure: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{5} + a^{4} + 3 a^{3} - 4 a^{2} + 2 : 4 a^{5} - 5 a^{4} - 12 a^{3} + 19 a^{2} - 7 : 1\right)$
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: \( 45606.094824939202644530546191781771836 \)
Tamagawa product: \( 2 \)
Torsion order: \(10\)
Leading coefficient: \( 1.49691 \)
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\((j)_{-}\)
\((-a^4+a^3+3a^2-3a-2)\) \(79\) \(2\) \(I_{2}\) Non-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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 79.4-c consists of curves linked by isogenies of degrees dividing 10.

Base change

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

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