Properties

 Label 6.6.371293.1-1.1-a1 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$1$$ CM no Base change yes Q-curve yes Torsion order $$1$$ 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

$${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-8a+1\right){x}^{2}+\left(-6649a^{5}+3526a^{4}+36471a^{3}-7581a^{2}-48724a-11069\right){x}-549202a^{5}+227195a^{4}+2939749a^{3}-400010a^{2}-3734843a-839159$$
sage: E = EllipticCurve([K([-2,2,1,-4,0,1]),K([1,-8,0,6,0,-1]),K([-1,-2,1,1,0,0]),K([-11069,-48724,-7581,36471,3526,-6649]),K([-839159,-3734843,-400010,2939749,227195,-549202])])

gp: E = ellinit([Polrev([-2,2,1,-4,0,1]),Polrev([1,-8,0,6,0,-1]),Polrev([-1,-2,1,1,0,0]),Polrev([-11069,-48724,-7581,36471,3526,-6649]),Polrev([-839159,-3734843,-400010,2939749,227195,-549202])], K);

magma: E := EllipticCurve([K![-2,2,1,-4,0,1],K![1,-8,0,6,0,-1],K![-1,-2,1,1,0,0],K![-11069,-48724,-7581,36471,3526,-6649],K![-839159,-3734843,-400010,2939749,227195,-549202]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(1)$$ = $$(1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1$$ = 1 sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(1)$$ = $$(1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1$$ = 1 sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-3387888351672962316333 a^{4} + 3387888351672962316333 a^{3} + 13551553406691849265332 a^{2} - 6775776703345924632666 a - 14577323462934449612494$$ 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: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.0026716469568848618763422560119573014187$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.571393$$ Analytic order of Ш: $$130321$$ (rounded)

Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$3$$ 3B
$$19$$ 19B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 19 and 57.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 57.

Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following elliptic curve:

Base field Curve
$$\Q(\sqrt{13})$$ 2.2.13.1-169.1-a1