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Properties

Label 6.6.300125.1-1.1-a2
Base field 6.6.300125.1
Conductor norm \( 1 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 37 \)
Rank \( 0 \)

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Base field 6.6.300125.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-2a^{5}+a^{4}+14a^{3}+4a^{2}-9a-2\right){x}{y}+\left(-5a^{5}+a^{4}+37a^{3}+18a^{2}-26a-7\right){y}={x}^{3}+\left(-3a^{5}+2a^{4}+20a^{3}+3a^{2}-12a\right){x}^{2}+\left(7a^{5}-a^{4}-51a^{3}-28a^{2}+27a+11\right){x}-13a^{5}+4a^{4}+92a^{3}+40a^{2}-56a-16\)
sage: E = EllipticCurve([K([-2,-9,4,14,1,-2]),K([0,-12,3,20,2,-3]),K([-7,-26,18,37,1,-5]),K([11,27,-28,-51,-1,7]),K([-16,-56,40,92,4,-13])])
 
gp: E = ellinit([Polrev([-2,-9,4,14,1,-2]),Polrev([0,-12,3,20,2,-3]),Polrev([-7,-26,18,37,1,-5]),Polrev([11,27,-28,-51,-1,7]),Polrev([-16,-56,40,92,4,-13])], K);
 
magma: E := EllipticCurve([K![-2,-9,4,14,1,-2],K![0,-12,3,20,2,-3],K![-7,-26,18,37,1,-5],K![11,27,-28,-51,-1,7],K![-16,-56,40,92,4,-13]]);
 

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: \( -9317 \)
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: \(\Z/37\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : 6 a^{5} - a^{4} - 44 a^{3} - 23 a^{2} + 28 a + 9 : 1\right)$
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: \( 391404.46794104455867972617384800786307 \)
Tamagawa product: \( 1 \)
Torsion order: \(37\)
Leading coefficient: \( 0.521880 \)
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
\(37\) 37B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 37.
Its isogeny class 1.1-a consists of curves linked by isogenies of degree 37.

Base change

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

Base field Curve
\(\Q(\sqrt{5}) \) 2.2.5.1-2401.1-c2

Additional information

This elliptic curve has everywhere good reduction and a rational point of order $37$.