# Properties

 Label 4.4.1125.1-31.4-b2 Base field $$\Q(\zeta_{15})^+$$ Conductor $$(2a^3-8a+1)$$ Conductor norm $$31$$ CM no Base change no Q-curve no Torsion order $$8$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{15})^+$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -4, -1, 1]))

gp: K = nfinit(Pol(Vecrev([1, 4, -4, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);

## Weierstrass equation

$${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}-4{x}-4a^{3}-4a^{2}+11a+10$$
sage: E = EllipticCurve([K([-2,-3,1,1]),K([-3,-3,1,1]),K([0,0,0,0]),K([-4,0,0,0]),K([10,11,-4,-4])])

gp: E = ellinit([Pol(Vecrev([-2,-3,1,1])),Pol(Vecrev([-3,-3,1,1])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-4,0,0,0])),Pol(Vecrev([10,11,-4,-4]))], K);

magma: E := EllipticCurve([K![-2,-3,1,1],K![-3,-3,1,1],K![0,0,0,0],K![-4,0,0,0],K![10,11,-4,-4]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(2a^3-8a+1)$$ = $$(2a^3-8a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$31$$ = $$31$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(a^3+26a^2-3a-66)$$ = $$(2a^3-8a+1)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$923521$$ = $$31^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{26935491035136}{923521} a^{3} - \frac{9133676668039}{923521} a^{2} + \frac{95674789534220}{923521} a + \frac{20462261904636}{923521}$$ 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: $$\Z/2\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-2 : a^{3} + a^{2} - 3 a - 2 : 1\right)$ $\left(a^{3} + a^{2} - 3 a + 2 : 2 a^{3} + 2 a^{2} - 6 a + 2 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$757.613584259798$$ Tamagawa product: $$2$$ Torsion order: $$8$$ Leading coefficient: $$0.705864781284238$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$(2a^3-8a+1)$$ $$31$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 31.4-b consists of curves linked by isogenies of degrees dividing 8.

## Base change

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