# Properties

 Base field 6.6.1241125.1 Label 6.6.1241125.1-25.2-b2 Conductor $$(5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)$$ Conductor norm $$25$$ CM no base-change no Q-curve no Torsion order $$5$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 6.6.1241125.1

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

$$y^2 + \left(-a^{5} + 7 a^{3} + 2 a^{2} - 11 a - 6\right) x y + \left(-5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 52 a - 19\right) y = x^{3} + \left(-3 a^{5} + a^{4} + 20 a^{3} - 30 a - 12\right) x^{2} + \left(-5 a^{5} + a^{4} + 36 a^{3} + 3 a^{2} - 62 a - 25\right) x - 6 a^{5} + a^{4} + 41 a^{3} + 4 a^{2} - 64 a - 27$$
sage: E = EllipticCurve(K, [-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27])

gp: E = ellinit([-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27],K)

magma: E := ChangeRing(EllipticCurve([-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)$$ = $$\left(5, a - 2\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$25$$ = $$5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(625,125 a^{5} - 875 a^{3} - 125 a^{2} + 1375 a + 750,116 a^{5} + a^{4} - 813 a^{3} - 120 a^{2} + 1279 a + 518,65 a^{5} + a^{4} - 455 a^{3} - 69 a^{2} + 714 a + 317,78 a^{5} - 546 a^{3} - 78 a^{2} + 859 a + 511,122 a^{5} - 854 a^{3} - 121 a^{2} + 1342 a + 553)$$ = $$\left(5, a - 2\right)^{7}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$78125$$ = $$5^{7}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{1332544}{5} a^{5} + \frac{2260678}{5} a^{4} - \frac{5421687}{5} a^{3} - \frac{11771837}{5} a^{2} - 1128428 a - \frac{732232}{5}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(0 : 3 a^{5} - a^{4} - 21 a^{3} + 34 a + 13 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(5, a - 2\right)$$ $$5$$ $$2$$ $$I_{1}^*$$ Additive $$1$$ $$2$$ $$7$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$3$$ 3B
$$5$$ 5B.1.1[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5 and 15.
Its isogeny class 25.2-b consists of curves linked by isogenies of degrees dividing 15.

## Base change

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