# Properties

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

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field6.6.300125.1

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(Pol(Vecrev([-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([Pol(Vecrev([-2,-9,4,14,1,-2])),Pol(Vecrev([0,-12,3,20,2,-3])),Pol(Vecrev([-7,-26,18,37,1,-5])),Pol(Vecrev([11,27,-28,-51,-1,7])),Pol(Vecrev([-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: not available sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: not available Tamagawa product: $$1$$ Torsion order: $$37$$ Leading coefficient: not available Analytic order of Ш: not available

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

## 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 curve is not the base change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.

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