# Properties

 Label 6.6.371293.1-79.3-d3 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$79$$ CM no Base change no Q-curve no Torsion order $$4$$ 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

$${y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+3\right){x}^{2}+\left(-4a^{5}-2a^{4}+17a^{3}+9a^{2}-10a+1\right){x}+12a^{5}+7a^{4}-51a^{3}-33a^{2}+29a+11$$
sage: E = EllipticCurve([K([1,5,0,-5,0,1]),K([3,-1,-4,4,1,-1]),K([0,1,0,0,0,0]),K([1,-10,9,17,-2,-4]),K([11,29,-33,-51,7,12])])

gp: E = ellinit([Polrev([1,5,0,-5,0,1]),Polrev([3,-1,-4,4,1,-1]),Polrev([0,1,0,0,0,0]),Polrev([1,-10,9,17,-2,-4]),Polrev([11,29,-33,-51,7,12])], K);

magma: E := EllipticCurve([K![1,5,0,-5,0,1],K![3,-1,-4,4,1,-1],K![0,1,0,0,0,0],K![1,-10,9,17,-2,-4],K![11,29,-33,-51,7,12]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^5-4a^3+3a-2)$$ = $$(a^5-4a^3+3a-2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$79$$ = $$79$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(3a^5+a^4-21a^3-2a^2+27a+1)$$ = $$(a^5-4a^3+3a-2)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-493039$$ = $$-79^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1576208327952}{493039} a^{5} - \frac{546637380727}{493039} a^{4} - \frac{8080139654401}{493039} a^{3} + \frac{801868607748}{493039} a^{2} + \frac{9414033518410}{493039} a + \frac{2112843364295}{493039}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-2 a^{5} - a^{4} + 9 a^{3} + 5 a^{2} - 7 a - 2 : 6 a^{5} + 3 a^{4} - 26 a^{3} - 15 a^{2} + 16 a + 6 : 1\right)$ 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: $$12962.521508497463063416447407328350439$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$1.32957$$ 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$$(j)_{-}$$
$$(a^5-4a^3+3a-2)$$ $$79$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$

## 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

## 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 79.3-d consists of curves linked by isogenies of degrees dividing 12.

## Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.