# Properties

 Label 6.6.1241125.1-25.2-a1 Base field 6.6.1241125.1 Conductor $$(-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)$$ Conductor norm $$25$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field6.6.1241125.1

Generator $$a$$, with minimal polynomial $$x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 7*x^4 - 2*x^3 + 11*x^2 + 7*x + 1)

gp: K = nfinit(a^6 - 7*a^4 - 2*a^3 + 11*a^2 + 7*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);

## Weierstrass equation

$$y^2+\left(-a^{5}+7a^{3}+2a^{2}-10a-6\right)xy+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-53a-19\right)y=x^{3}+\left(2a^{5}-a^{4}-13a^{3}+2a^{2}+20a+5\right)x^{2}+\left(-6a^{5}+2a^{4}+40a^{3}-60a-21\right)x-5a^{5}+a^{4}+35a^{3}+3a^{2}-55a-24$$
sage: E = EllipticCurve(K, [-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24])

gp: E = ellinit([-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24],K)

magma: E := ChangeRing(EllipticCurve([-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)$$ = $$\left(5, a - 2\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$25$$ = $$5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(-3 a^{5} + a^{4} + 21 a^{3} - a^{2} - 35 a - 9)$$ = $$\left(5, a - 2\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$125$$ = $$5^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-2708 a^{5} + 733 a^{4} + 12894 a^{3} - 3948 a^{2} - 9886 a - 1842$$ 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 $\left(-2 a^{5} + a^{4} + 13 a^{3} - 2 a^{2} - 20 a - 5 : -a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 9 a + 1 : 1\right)$ Height $$0.00896445338049678$$ 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: not available sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.00896445338049678$$ Period: not available Tamagawa product: $$2$$ Torsion order: $$1$$ 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$$(j)_{-}$$
$$\left(5, a - 2\right)$$ $$5$$ $$2$$ $$III$$ Additive $$-1$$ $$2$$ $$3$$ $$0$$

## 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 25.2-a consists of this curve only.

## Base change

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