Properties

 Label 3.3.49.1-71.3-a2 Base field $$\Q(\zeta_{7})^+$$ Conductor $$(a^{2} - 6)$$ Conductor norm $$71$$ CM no Base change no Q-curve no Torsion order $$10$$ Rank $$0$$

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Base field$$\Q(\zeta_{7})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 2 x + 1$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)

gp: K = nfinit(a^3 - a^2 - 2*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);

Weierstrass equation

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

gp: E = ellinit([1, a^2 - 2, a^2 - 1, -a^2 + 1, 0],K)

magma: E := ChangeRing(EllipticCurve([1, a^2 - 2, a^2 - 1, -a^2 + 1, 0]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(a^{2} - 6)$$ = $$\left(a^{2} - 6\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$71$$ = $$71$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(a^{2} - 6)$$ = $$\left(a^{2} - 6\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$71$$ = $$71$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2121}{71} a^{2} - \frac{713073}{71} a + \frac{1242209}{71}$$ 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: $\left(0 : -a^{2} + 1 : 1\right)$ 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: $$411.200909713333$$ Tamagawa product: $$1$$ Torsion order: $$10$$ Leading coefficient: $$0.587429871019048$$ 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$$(j)_{-}$$
$$\left(a^{2} - 6\right)$$ $$71$$ $$1$$ $$I_{1}$$ 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 71.3-a consists of curves linked by isogenies of degrees dividing 10.

Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.