# Properties

 Label 6.6.371293.1-79.4-c2 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$79$$ CM no Base change no Q-curve no Torsion order $$10$$ 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

$${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(a^{4}+a^{3}-3a^{2}-4a\right){x}-2a^{5}+a^{4}+12a^{3}-a^{2}-16a-5$$
sage: E = EllipticCurve([K([0,-2,-3,1,1,0]),K([2,3,-1,-1,0,0]),K([-2,2,1,-4,0,1]),K([0,-4,-3,1,1,0]),K([-5,-16,-1,12,1,-2])])

gp: E = ellinit([Polrev([0,-2,-3,1,1,0]),Polrev([2,3,-1,-1,0,0]),Polrev([-2,2,1,-4,0,1]),Polrev([0,-4,-3,1,1,0]),Polrev([-5,-16,-1,12,1,-2])], K);

magma: E := EllipticCurve([K![0,-2,-3,1,1,0],K![2,3,-1,-1,0,0],K![-2,2,1,-4,0,1],K![0,-4,-3,1,1,0],K![-5,-16,-1,12,1,-2]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a^4+a^3+3a^2-3a-2)$$ = $$(-a^4+a^3+3a^2-3a-2)$$ 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: $$(2a^5-a^4-9a^3+3a^2+7a-1)$$ = $$(-a^4+a^3+3a^2-3a-2)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-79$$ = $$-79$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{9875329}{79} a^{5} - \frac{4876294}{79} a^{4} + 531385 a^{3} + \frac{23192531}{79} a^{2} - \frac{24136997}{79} a - \frac{6376947}{79}$$ 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: $$\Z/10\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-a^{5} + 5 a^{3} + a^{2} - 6 a - 2 : -a^{5} + 5 a^{3} - 5 a : 1\right)$ 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: $$91212.189649878405289061092383563543672$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$1.49691$$ 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$$(j)_{-}$$
$$(-a^4+a^3+3a^2-3a-2)$$ $$79$$ $$1$$ $$I_{1}$$ 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
$$2$$ 2B
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 79.4-c consists of curves linked by isogenies of degrees dividing 10.

## Base change

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

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