# Properties

 Label 6.6.1241125.1-25.2-c1 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 $$0$$

# 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

$$y^2+\left(-2a^{5}+a^{4}+13a^{3}-2a^{2}-18a-6\right)xy+\left(-a^{5}+7a^{3}+2a^{2}-11a-5\right)y=x^{3}+\left(-a^{2}+a+2\right)x^{2}+\left(26a^{5}-12a^{4}-176a^{3}+29a^{2}+270a+59\right)x-51a^{5}+24a^{4}+346a^{3}-61a^{2}-534a-106$$
sage: E = EllipticCurve(K, [-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106])

gp: E = ellinit([-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106],K)

magma: E := ChangeRing(EllipticCurve([-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106]),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: $$(-14 a^{5} + 5 a^{4} + 162 a^{3} - 23 a^{2} - 282 a + 33)$$ = $$\left(5, a - 2\right)^{19}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$19073486328125$$ = $$5^{19}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1376405109}{78125} a^{5} - \frac{1325860482}{78125} a^{4} - \frac{7597745332}{78125} a^{3} + \frac{2698464798}{78125} a^{2} + \frac{2247897599}{15625} a + \frac{2321410818}{78125}$$ 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: 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: $$0$$ Regulator: $$1$$ 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$$(j)_{-}$$
$$\left(5, a - 2\right)$$ $$5$$ $$2$$ $$I_{13}^*$$ Additive $$1$$ $$2$$ $$19$$ $$13$$

## 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-c 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.