# Properties

 Label 6.6.371293.1-64.1-b2 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$64$$ CM no Base change yes Q-curve yes Torsion order $$7$$ Rank $$1$$

# 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

$${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+2\right){x}^{2}+\left(a^{5}+2a^{4}-a^{3}-a^{2}+3a+1\right){x}-20a^{5}-7a^{4}+93a^{3}+50a^{2}-52a-14$$
sage: E = EllipticCurve([K([0,3,-3,-4,1,1]),K([2,5,-4,-5,1,1]),K([1,5,-3,-5,1,1]),K([1,3,-1,-1,2,1]),K([-14,-52,50,93,-7,-20])])

gp: E = ellinit([Polrev([0,3,-3,-4,1,1]),Polrev([2,5,-4,-5,1,1]),Polrev([1,5,-3,-5,1,1]),Polrev([1,3,-1,-1,2,1]),Polrev([-14,-52,50,93,-7,-20])], K);

magma: E := EllipticCurve([K![0,3,-3,-4,1,1],K![2,5,-4,-5,1,1],K![1,5,-3,-5,1,1],K![1,3,-1,-1,2,1],K![-14,-52,50,93,-7,-20]]);

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: $$(-4)$$ = $$(2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4096$$ = $$64^{2}$$ 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: $$1$$ Generator $\left(-a^{4} + 3 a^{2} - a : 2 a^{5} + 3 a^{4} - 7 a^{3} - 10 a^{2} + 2 a + 2 : 1\right)$ Height $$0.16981435127840786675757737391447202240$$ 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: $\left(-a^{2} - a + 1 : 3 a^{5} + 2 a^{4} - 11 a^{3} - 8 a^{2} + 5 a + 4 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.16981435127840786675757737391447202240$$ Period: $$36850.572209580007121685212569502764530$$ Tamagawa product: $$2$$ Torsion order: $$7$$ Leading coefficient: $$2.51505$$ 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$$(j)_{-}$$
$$(2)$$ $$64$$ $$2$$ $$I_{2}$$ 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 64.1-b 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 5 elliptic curves:

Base field Curve
$$\Q$$ 338.c2
$$\Q$$ 338.e2
$$\Q(\sqrt{13})$$ 2.2.13.1-676.1-b2
3.3.169.1 a curve with conductor norm 1352 (not in the database)
3.3.169.1 3.3.169.1-8.1-a2