# Properties

 Label 3.3.316.1-4.3-a1 Base field 3.3.316.1 Conductor $$(-a-1)$$ Conductor norm $$4$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$0$$

# 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

$${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-2770a^{2}+1478a+11792\right){x}+1679357a^{2}-888891a-7135787$$
sage: E = EllipticCurve([K([-2,1,1]),K([-3,1,1]),K([1,1,0]),K([11792,1478,-2770]),K([-7135787,-888891,1679357])])

gp: E = ellinit([Pol(Vecrev([-2,1,1])),Pol(Vecrev([-3,1,1])),Pol(Vecrev([1,1,0])),Pol(Vecrev([11792,1478,-2770])),Pol(Vecrev([-7135787,-888891,1679357]))], K);

magma: E := EllipticCurve([K![-2,1,1],K![-3,1,1],K![1,1,0],K![11792,1478,-2770],K![-7135787,-888891,1679357]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a-1)$$ = $$(-a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$4$$ = $$2^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2a^2+a+7)$$ = $$(-a+1)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$16$$ = $$2^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$342336 a^{2} + 460416 a - 292544$$ 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: $\left(-41 a^{2} + \frac{41}{2} a + \frac{345}{2} : -\frac{99}{4} a^{2} + \frac{63}{4} a + \frac{221}{2} : 1\right)$ 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: $$98.1032954784847$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$1.37968538479767$$ 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$$(j)_{-}$$
$$(-a+1)$$ $$2$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$

## 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.
Its isogeny class 4.3-a consists of curves linked by isogenies of degree 2.

## Base change

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