Properties

 Label 6.6.371293.1-79.3-d7 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$79$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$0$$

Related objects

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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

$${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a-3\right){x}^{2}+\left(-56a^{5}+144a^{4}+90a^{3}-446a^{2}+276a-42\right){x}-1161a^{5}+2386a^{4}+3200a^{3}-7763a^{2}+1278a+1453$$
sage: E = EllipticCurve([K([1,-2,0,1,0,0]),K([-3,-4,4,1,-1,0]),K([1,-3,-3,1,1,0]),K([-42,276,-446,90,144,-56]),K([1453,1278,-7763,3200,2386,-1161])])

gp: E = ellinit([Polrev([1,-2,0,1,0,0]),Polrev([-3,-4,4,1,-1,0]),Polrev([1,-3,-3,1,1,0]),Polrev([-42,276,-446,90,144,-56]),Polrev([1453,1278,-7763,3200,2386,-1161])], K);

magma: E := EllipticCurve([K![1,-2,0,1,0,0],K![-3,-4,4,1,-1,0],K![1,-3,-3,1,1,0],K![-42,276,-446,90,144,-56],K![1453,1278,-7763,3200,2386,-1161]]);

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: $$(-a^5+4a^3-3a+2)$$ = $$(a^5-4a^3+3a-2)$$ 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: $$-\frac{41494566893660714555288080}{79} a^{5} + \frac{51497801567134241086634054}{79} a^{4} + \frac{195058086408065742590919756}{79} a^{3} - \frac{213001575932755291713651877}{79} a^{2} - \frac{197618395730591263439326644}{79} a + \frac{172124231458098209062968630}{79}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-\frac{17}{4} a^{5} + \frac{19}{4} a^{4} + \frac{35}{2} a^{3} - \frac{35}{2} a^{2} - \frac{31}{4} a + \frac{13}{2} : \frac{13}{4} a^{5} - \frac{51}{8} a^{4} - \frac{93}{8} a^{3} + \frac{157}{8} a^{2} + \frac{15}{4} a - \frac{51}{8} : 1\right)$ 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: $$202.53939857027286036588199073950547562$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$1.32957$$ Analytic order of Ш: $$16$$ (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$$(j)_{-}$$
$$(a^5-4a^3+3a-2)$$ $$79$$ $$1$$ $$I_{1}$$ 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
$$3$$ 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.