Properties

 Label 3.3.361.1-512.1-b3 Base field 3.3.361.1 Conductor $$(8)$$ Conductor norm $$512$$ CM no Base change yes: 2888.e1 Q-curve yes Torsion order $$4$$ Rank $$1$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field3.3.361.1

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 6 x + 7$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -6, -1, 1]))

gp: K = nfinit(Pol(Vecrev([7, -6, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -6, -1, 1]);

Weierstrass equation

$${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(112a^{2}-23a-759\right){x}-1232a^{2}+22a+7799$$
sage: E = EllipticCurve([K([0,0,0]),K([1,-1,0]),K([0,0,0]),K([-759,-23,112]),K([7799,22,-1232])])

gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([1,-1,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([-759,-23,112])),Pol(Vecrev([7799,22,-1232]))], K);

magma: E := EllipticCurve([K![0,0,0],K![1,-1,0],K![0,0,0],K![-759,-23,112],K![7799,22,-1232]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(8)$$ = $$(2)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$512$$ = $$8^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(16)$$ = $$(2)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4096$$ = $$8^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$1462911232$$ 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: $$1$$ Generator $\left(-\frac{47}{49} a^{2} + \frac{137}{49} a + \frac{88}{7} : \frac{1744}{343} a^{2} + \frac{2305}{343} a - \frac{740}{49} : 1\right)$ Height $$0.861382156778280$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(9 a^{2} + 4 a - 47 : 0 : 1\right)$ $\left(-2 a^{2} + 2 a + 15 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.861382156778280$$ Period: $$192.234344150077$$ Tamagawa product: $$2$$ Torsion order: $$4$$ Leading coefficient: $$3.26816909152996$$ 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$$(j)_{-}$$
$$(2)$$ $$8$$ $$2$$ $$III$$ Additive $$1$$ $$3$$ $$4$$ $$0$$

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
$$5$$ 5S4

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 512.1-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 2888.e1, defined over $$\Q$$, so it is also a $$\Q$$-curve.