# Properties

 Label 4.4.1125.1-31.4-b6 Base field $$\Q(\zeta_{15})^+$$ Conductor $$(2a^3-8a+1)$$ Conductor norm $$31$$ CM no Base change no Q-curve no Torsion order $$2$$ 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

$${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-340a^{3}-335a^{2}+945a+131\right){x}-5083a^{3}-4753a^{2}+13402a+2742$$
sage: E = EllipticCurve([K([-2,-3,1,1]),K([-3,-3,1,1]),K([0,0,0,0]),K([131,945,-335,-340]),K([2742,13402,-4753,-5083])])

gp: E = ellinit([Pol(Vecrev([-2,-3,1,1])),Pol(Vecrev([-3,-3,1,1])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([131,945,-335,-340])),Pol(Vecrev([2742,13402,-4753,-5083]))], K);

magma: E := EllipticCurve([K![-2,-3,1,1],K![-3,-3,1,1],K![0,0,0,0],K![131,945,-335,-340],K![2742,13402,-4753,-5083]]);

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: $$\frac{19386841577544688278603898209}{923521} a^{3} + \frac{16034680544860826249617094257}{923521} a^{2} - \frac{48250547157163713883980478227}{923521} a - \frac{10610770058214958430895584744}{923521}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-5 a^{3} - 5 a^{2} + \frac{59}{4} a - \frac{3}{2} : 6 a^{3} + \frac{47}{8} a^{2} - \frac{31}{2} a - \frac{53}{8} : 1\right)$ 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: $$2.95942806351483$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.705864781284238$$ Analytic order of Ш: $$16$$ (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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
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.