# Properties

 Label 6.6.371293.1-64.1-a5 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$64$$ CM no Base change yes Q-curve yes Torsion order $$5$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\zeta_{13})^+$$

Generator $$a$$, with minimal polynomial $$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))

gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);

## Weierstrass equation

$${y}^2+{x}{y}+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){y}={x}^{3}+{x}^{2}+\left(a^{4}-2a^{3}-4a^{2}+4a\right){x}-2a^{4}+7a^{2}-3$$
sage: E = EllipticCurve([K([1,0,0,0,0,0]),K([1,0,0,0,0,0]),K([3,-2,-4,1,1,0]),K([0,4,-4,-2,1,0]),K([-3,0,7,0,-2,0])])

gp: E = ellinit([Polrev([1,0,0,0,0,0]),Polrev([1,0,0,0,0,0]),Polrev([3,-2,-4,1,1,0]),Polrev([0,4,-4,-2,1,0]),Polrev([-3,0,7,0,-2,0])], K);

magma: E := EllipticCurve([K![1,0,0,0,0,0],K![1,0,0,0,0,0],K![3,-2,-4,1,1,0],K![0,4,-4,-2,1,0],K![-3,0,7,0,-2,0]]);

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: $$64$$ = $$64$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2)$$ = $$(2)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$64$$ = $$64$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{461373}{2} a^{4} - \frac{461373}{2} a^{3} - 922746 a^{2} + 461373 a + \frac{321323}{2}$$ 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: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -a^{3} + 2 a : 1\right)$ 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: $$23287.107506209544771182610817319643780$$ Tamagawa product: $$1$$ Torsion order: $$5$$ Leading coefficient: $$1.52868$$ 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$$(j)_{-}$$
$$(2)$$ $$64$$ $$1$$ $$I_{1}$$ 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
$$3$$ 3B
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5, 9, 15 and 45.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 45.

## Base change

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

Base field Curve
$$\Q(\sqrt{13})$$ 2.2.13.1-4.1-a4