# Properties

 Label 3.3.49.1-64.1-a3 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(4)$$ Conductor norm $$64$$ CM no Base change yes: 196.b1 Q-curve yes Torsion order $$4$$ 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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -1, 1]))

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

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

## Weierstrass equation

$${y}^2={x}^{3}-a{x}^{2}+\left(102a^{2}-61a-244\right){x}+640a^{2}-360a-1460$$
sage: E = EllipticCurve([K([0,0,0]),K([0,-1,0]),K([0,0,0]),K([-244,-61,102]),K([-1460,-360,640])])

gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([0,-1,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([-244,-61,102])),Pol(Vecrev([-1460,-360,640]))], K);

magma: E := EllipticCurve([K![0,0,0],K![0,-1,0],K![0,0,0],K![-244,-61,102],K![-1460,-360,640]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(4)$$ = $$(2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$64$$ = $$8^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(16)$$ = $$(2)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4096$$ = $$8^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$406749952$$ 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: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(5 a^{2} - 10 : 0 : 1\right)$ $\left(4 a^{2} - 4 a - 8 : 0 : 1\right)$ 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: $$6.98663268068574$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$0.561425840412247$$ Analytic order of Ш: $$9$$ (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$$(j)_{-}$$
$$(2)$$ $$8$$ $$1$$ $$IV$$ 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
$$2$$ 2Cs
$$3$$ 3B.1.2

## Isogenies and isogeny class

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

## Base change

This curve is the base change of elliptic curves 196.b1, defined over $$\Q$$, so it is also a $$\Q$$-curve.