# Properties

 Label 6.6.371293.1-79.3-d6 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

$${y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+3\right){x}^{2}+\left(41a^{5}+18a^{4}-173a^{3}-86a^{2}+95a+16\right){x}+259a^{5}+137a^{4}-1094a^{3}-644a^{2}+601a+171$$
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([16,95,-86,-173,18,41]),K([171,601,-644,-1094,137,259])])

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([16,95,-86,-173,18,41]),Polrev([171,601,-644,-1094,137,259])], 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![16,95,-86,-173,18,41],K![171,601,-644,-1094,137,259]]);

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: $$(49a^5-22a^4-190a^3+69a^2+53a-70)$$ = $$(a^5-4a^3+3a-2)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$243087455521$$ = $$79^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{9864017269525778498272880}{243087455521} a^{5} - \frac{2868359809586230613384259}{243087455521} a^{4} - \frac{51354355161968675541349721}{243087455521} a^{3} + \frac{3035058454072124864303972}{243087455521} a^{2} + \frac{61336596742421539337407014}{243087455521} a + \frac{13908462111951939654564430}{243087455521}$$ 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: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(4 a^{5} - a^{4} - 18 a^{3} + 2 a^{2} + 14 a - 2 : a^{4} + a^{3} - 3 a^{2} - 4 a - 1 : 1\right)$ $\left(\frac{1}{2} a^{5} + a^{4} - \frac{5}{4} a^{3} - 4 a^{2} - \frac{9}{4} a + \frac{1}{4} : \frac{7}{4} a^{5} - \frac{69}{8} a^{3} - \frac{9}{8} a^{2} + \frac{67}{8} a + \frac{17}{8} : 1\right)$ 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: $$1620.3151885621828829270559259160438049$$ Tamagawa product: $$2$$ Torsion order: $$4$$ Leading coefficient: $$1.32957$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$(a^5-4a^3+3a-2)$$ $$79$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
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.