# Properties

 Label 3.3.316.1-32.1-c5 Base field 3.3.316.1 Conductor $$(-2a^2+8)$$ Conductor norm $$32$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field3.3.316.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))

gp: K = nfinit(Pol(Vecrev([2, -4, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);

## Weierstrass equation

$${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(486273921a^{2}-257393114a-2066246710\right){x}-8535646743838a^{2}+4518059339552a+36269162590988$$
sage: E = EllipticCurve([K([0,1,0]),K([-2,1,1]),K([-2,0,1]),K([-2066246710,-257393114,486273921]),K([36269162590988,4518059339552,-8535646743838])])

gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([-2,1,1])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([-2066246710,-257393114,486273921])),Pol(Vecrev([36269162590988,4518059339552,-8535646743838]))], K);

magma: E := EllipticCurve([K![0,1,0],K![-2,1,1],K![-2,0,1],K![-2066246710,-257393114,486273921],K![36269162590988,4518059339552,-8535646743838]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2a^2+8)$$ = $$(a)^{4}\cdot(-a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$32$$ = $$2^{4}\cdot2$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2a^2+12a+12)$$ = $$(a)^{12}\cdot(-a+1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-8192$$ = $$2^{12}\cdot2$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782$$ 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 $\left(-6131 a^{2} + 3293 a + 26138 : 1769 a^{2} - 318 a - 6396 : 1\right)$ Height $$0.264164657112299$$ 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: $\left(-\frac{24637}{4} a^{2} + 3255 a + 26162 : \frac{11613}{8} a^{2} - \frac{1525}{2} a - \frac{24633}{4} : 1\right)$ 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: $$0.264164657112299$$ Period: $$30.3125634354532$$ Tamagawa product: $$4$$  =  $$2^{2}\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$1.35137254263725$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_4^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$(-a+1)$$ $$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## 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, 8 and 16.
Its isogeny class 32.1-c consists of curves linked by isogenies of degrees dividing 16.

## Base change

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