# Properties

 Label 3.3.169.1-8.1-a2 Base field 3.3.169.1 Conductor norm $$8$$ CM no Base change yes Q-curve yes Torsion order $$7$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field3.3.169.1

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

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

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

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

## Weierstrass equation

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

gp: E = ellinit([Polrev([1,1,0]),Polrev([2,1,-1]),Polrev([-2,-1,1]),Polrev([3,0,-1]),Polrev([3,1,-1])], K);

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(2)$$ = $$(2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$8$$ = $$8$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-4)$$ = $$(2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-64$$ = $$-8^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{351}{4}$$ 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: $$\Z/7\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(0 : -1 : 1\right)$ 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: $$191.96502861089049706621044840443825023$$ Tamagawa product: $$2$$ Torsion order: $$7$$ Leading coefficient: $$0.60271594540310988089861993219603846227$$ 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$$(j)_{-}$$
$$(2)$$ $$8$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$7$$ 7B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 8.1-a consists of curves linked by isogenies of degree 7.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following elliptic curve:

Base field Curve
$$\Q$$ 338.e2