# Properties

 Label 4.4.1125.1-89.3-b1 Base field $$\Q(\zeta_{15})^+$$ Conductor norm $$89$$ CM no Base change no Q-curve no Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([1, 4, -4, -1, 1]));

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

## Weierstrass equation

$${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(82a^{3}+31a^{2}-291a-75\right){x}-396a^{3}-143a^{2}+1404a+326$$
sage: E = EllipticCurve([K([-2,1,1,0]),K([-2,4,1,-1]),K([0,-2,0,1]),K([-75,-291,31,82]),K([326,1404,-143,-396])])

gp: E = ellinit([Polrev([-2,1,1,0]),Polrev([-2,4,1,-1]),Polrev([0,-2,0,1]),Polrev([-75,-291,31,82]),Polrev([326,1404,-143,-396])], K);

magma: E := EllipticCurve([K![-2,1,1,0],K![-2,4,1,-1],K![0,-2,0,1],K![-75,-291,31,82],K![326,1404,-143,-396]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2a^3+a^2+7a-2)$$ = $$(-2a^3+a^2+7a-2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$89$$ = $$89$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(3a^3+6a^2-25a-27)$$ = $$(-2a^3+a^2+7a-2)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-704969$$ = $$-89^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{83630646617237}{704969} a^{3} - \frac{59371662388246}{704969} a^{2} + \frac{221021181736913}{704969} a + \frac{33436542823570}{704969}$$ 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: $$1$$ Generator $\left(7 a^{3} + 2 a^{2} - 25 a - 4 : 6 a^{3} + a^{2} - 22 a : 1\right)$ Height $$0.019242294063444979405768450589317675424$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-\frac{8}{3} a^{3} - \frac{5}{3} a^{2} + \frac{28}{3} a + \frac{14}{3} : \frac{22}{9} a^{3} + \frac{1}{3} a^{2} - \frac{86}{9} a + \frac{4}{9} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.019242294063444979405768450589317675424$$ Period: $$1474.0844125343780183372939428680433091$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$1.12756523308851$$ 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$$(j)_{-}$$
$$(-2a^3+a^2+7a-2)$$ $$89$$ $$3$$ $$I_{3}$$ 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
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 89.3-b consists of curves linked by isogenies of degree 3.

## Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.