# Properties

 Label 6.6.300125.1-71.5-a1 Base field 6.6.300125.1 Conductor $$(-7a^5+2a^4+50a^3+22a^2-30a-10)$$ Conductor norm $$71$$ CM no Base change no Q-curve no Torsion order $$2$$ 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

$${y}^2+\left(-6a^{5}+2a^{4}+43a^{3}+17a^{2}-29a-7\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-28a-10\right){y}={x}^{3}+\left(a^{5}-8a^{3}-4a^{2}+7a+2\right){x}^{2}+\left(108a^{5}-25a^{4}-776a^{3}-378a^{2}+466a+140\right){x}+536a^{5}-123a^{4}-3847a^{3}-1892a^{2}+2295a+696$$
sage: E = EllipticCurve([K([-7,-29,17,43,2,-6]),K([2,7,-4,-8,0,1]),K([-10,-28,23,44,1,-6]),K([140,466,-378,-776,-25,108]),K([696,2295,-1892,-3847,-123,536])])

gp: E = ellinit([Pol(Vecrev([-7,-29,17,43,2,-6])),Pol(Vecrev([2,7,-4,-8,0,1])),Pol(Vecrev([-10,-28,23,44,1,-6])),Pol(Vecrev([140,466,-378,-776,-25,108])),Pol(Vecrev([696,2295,-1892,-3847,-123,536]))], K);

magma: E := EllipticCurve([K![-7,-29,17,43,2,-6],K![2,7,-4,-8,0,1],K![-10,-28,23,44,1,-6],K![140,466,-378,-776,-25,108],K![696,2295,-1892,-3847,-123,536]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-7a^5+2a^4+50a^3+22a^2-30a-10)$$ = $$(-7a^5+2a^4+50a^3+22a^2-30a-10)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$71$$ = $$71$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(18a^5+5a^4-138a^3-130a^2+112a+68)$$ = $$(-7a^5+2a^4+50a^3+22a^2-30a-10)^{5}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-1804229351$$ = $$-71^{5}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{18671790400300000}{1804229351} a^{5} - \frac{5223136808131864}{1804229351} a^{4} - \frac{134463355085824784}{1804229351} a^{3} - \frac{59498874802185092}{1804229351} a^{2} + \frac{87859926083703700}{1804229351} a + \frac{25940020939985535}{1804229351}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-19 a^{5} + 4 a^{4} + 137 a^{3} + 69 a^{2} - 83 a - 27 : -9 a^{5} + 3 a^{4} + 64 a^{3} + 26 a^{2} - 41 a - 10 : 1\right)$ 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: $$2$$ Leading coefficient: not available Analytic order of Ш: not available

## 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$$(j)_{-}$$
$$(-7a^5+2a^4+50a^3+22a^2-30a-10)$$ $$71$$ $$1$$ $$I_{5}$$ Non-split multiplicative $$1$$ $$1$$ $$5$$ $$5$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 71.5-a consists of curves linked by isogenies of degree 2.

## Base change

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