Properties

Label 6.6.371293.1-79.4-b1
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
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

\({y}^2+{x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-3\right){x}^{2}+\left(4a^{5}+5a^{4}-19a^{3}-20a^{2}+14a+5\right){x}-29a^{5}-20a^{4}+127a^{3}+87a^{2}-82a-21\)
sage: E = EllipticCurve([K([1,0,0,0,0,0]),K([-3,2,1,-1,0,0]),K([2,3,-4,-4,1,1]),K([5,14,-20,-19,5,4]),K([-21,-82,87,127,-20,-29])])
 
gp: E = ellinit([Polrev([1,0,0,0,0,0]),Polrev([-3,2,1,-1,0,0]),Polrev([2,3,-4,-4,1,1]),Polrev([5,14,-20,-19,5,4]),Polrev([-21,-82,87,127,-20,-29])], K);
 
magma: E := EllipticCurve([K![1,0,0,0,0,0],K![-3,2,1,-1,0,0],K![2,3,-4,-4,1,1],K![5,14,-20,-19,5,4],K![-21,-82,87,127,-20,-29]]);
 

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: \( -\frac{36496668313}{6241} a^{5} + \frac{77850269004}{6241} a^{4} + \frac{1194357473}{79} a^{3} - \frac{253201872353}{6241} a^{2} + \frac{68224588757}{6241} a + \frac{32165821654}{6241} \)
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 $\left(a^{5} - 4 a^{3} - a^{2} + 2 a + 2 : 2 a^{5} + a^{4} - 9 a^{3} - 5 a^{2} + 6 a + 1 : 1\right)$
Height \(0.0011713697873235358072686064267420056249\)
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: \( 0.0011713697873235358072686064267420056249 \)
Period: \( 107610.22356721157453697223324609030866 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 2.48239 \)
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\((j)_{-}\)
\((-a^4+a^3+3a^2-3a-2)\) \(79\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 79.4-b consists of this curve only.

Base change

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

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