Properties

Label 6.6.371293.1-1.1-a3
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 1 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 19 \)
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

\({y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(-a^{5}+6a^{3}+a^{2}-8a-1\right){x}^{2}+\left(2a^{5}-9a^{3}+9a+3\right){x}+a^{4}+a^{3}-2a^{2}-3a-1\)
sage: E = EllipticCurve([K([-1,-3,1,1,0,0]),K([-1,-8,1,6,0,-1]),K([3,5,-4,-5,1,1]),K([3,9,0,-9,0,2]),K([-1,-3,-2,1,1,0])])
 
gp: E = ellinit([Polrev([-1,-3,1,1,0,0]),Polrev([-1,-8,1,6,0,-1]),Polrev([3,5,-4,-5,1,1]),Polrev([3,9,0,-9,0,2]),Polrev([-1,-3,-2,1,1,0])], K);
 
magma: E := EllipticCurve([K![-1,-3,1,1,0,0],K![-1,-8,1,6,0,-1],K![3,5,-4,-5,1,1],K![3,9,0,-9,0,2],K![-1,-3,-2,1,1,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 17787 a^{4} - 17787 a^{3} - 71148 a^{2} + 35574 a + 13586 \)
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: \(\Z/19\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{3} + 2 a : a^{5} - a^{4} - 2 a^{3} + 3 a^{2} - 1 : 1\right)$
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: \( 125689.98480761734253583449161007777963 \)
Tamagawa product: \( 1 \)
Torsion order: \(19\)
Leading coefficient: \( 0.571393 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B
\(19\) 19B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 19 and 57.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 57.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q(\sqrt{13}) \) 2.2.13.1-169.1-a4