# Properties

 Label 3.3.49.1-64.1-a8 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(4)$$ Conductor norm $$64$$ CM no Base change no Q-curve yes Torsion order $$6$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{7})^+$$

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

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)

gp: K = nfinit(a^3 - a^2 - 2*a + 1);

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

## Weierstrass equation

$$y^2=x^{3}-ax^{2}+\left(-8a^{2}+4a+16\right)x+4a^{2}-8$$
sage: E = EllipticCurve(K, [0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8])

gp: E = ellinit([0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8],K)

magma: E := ChangeRing(EllipticCurve([0, -a, 0, -8*a^2 + 4*a + 16, 4*a^2 - 8]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(4)$$ = $$\left(2\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$64$$ = $$8^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(256)$$ = $$\left(2\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$16777216$$ = $$8^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$143392 a^{2} - 208912 a + 66240$$ 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-4 a^{2} + 4 a + 10 : -18 a^{2} + 8 a + 38 : 1\right)$ 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: $$47.1597705946287$$ Tamagawa product: $$3$$ Torsion order: $$6$$ Leading coefficient: $$0.561425840412247$$ 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$$(j)_{-}$$
$$\left(2\right)$$ $$8$$ $$3$$ $$IV^*$$ Additive $$-1$$ $$2$$ $$8$$ $$0$$

## 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
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 12.

## Base change

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