# Properties

 Base field 4.4.10512.1 Label 4.4.10512.1-4.1-d6 Conductor $$(2,a^{3} - a^{2} - 5 a - 2)$$ Conductor norm $$4$$ 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 4.4.10512.1

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

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);

## Weierstrass equation

$$y^2 + a x y + y = x^{3} + \left(a^{2} - 2 a - 5\right) x^{2} + \left(-a^{3} + 5 a + 5\right) x + a + 1$$
sage: E = EllipticCurve(K, [a, a^2 - 2*a - 5, 1, -a^3 + 5*a + 5, a + 1])

gp: E = ellinit([a, a^2 - 2*a - 5, 1, -a^3 + 5*a + 5, a + 1],K)

magma: E := ChangeRing(EllipticCurve([a, a^2 - 2*a - 5, 1, -a^3 + 5*a + 5, a + 1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2,a^{3} - a^{2} - 5 a - 2)$$ = $$\left(a^{3} - a^{2} - 5 a\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$4$$ = $$4$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(4,2 a^{3} - 2 a^{2} - 10 a,4 a,2 a^{2} - 2 a - 6)$$ = $$\left(a^{3} - a^{2} - 5 a\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$64$$ = $$4^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{580191}{4} a^{3} + \frac{580191}{4} a^{2} + \frac{2900955}{4} a - 104276$$ 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); $\left(-a^{3} + a^{2} + 5 a + 3 : a^{3} - 2 a^{2} - 3 a + 1 : 1\right)$ 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$$(j)_{-}$$
$$\left(a^{3} - a^{2} - 5 a\right)$$ $$4$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5 and 15.
Its isogeny class 4.1-d consists of curves linked by isogenies of degrees dividing 45.

## Base change

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