Properties

 Label 2.0.3.1-126976.2-e2 Base field $$\Q(\sqrt{-3})$$ Conductor $$(384a-320)$$ Conductor norm $$126976$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-3})$$

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

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

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

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

Weierstrass equation

$${y}^2={x}^{3}-{x}^{2}+\left(32a-33\right){x}-32a+65$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-33,32]),K([65,-32])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-33,32])),Pol(Vecrev([65,-32]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![-33,32],K![65,-32]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(384a-320)$$ = $$(2)^{6}\cdot(6a-5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$126976$$ = $$4^{6}\cdot31$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(524288a-3145728)$$ = $$(2)^{19}\cdot(6a-5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$8521215115264$$ = $$4^{19}\cdot31$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{24551}{62} a + \frac{66955}{62}$$ 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 $\left(-8 a + 13 : 32 a - 32 : 1\right)$ Height $$0.350530872893221$$ 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: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.350530872893221$$ Period: $$1.15134933143096$$ Tamagawa product: $$4$$  =  $$2^{2}\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$3.72814454992109$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$4$$ $$I_{9}^{*}$$ Additive $$1$$ $$6$$ $$19$$ $$1$$
$$(6a-5)$$ $$31$$ $$1$$ $$I_{1}$$ 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
$$5$$ 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5 and 25.
Its isogeny class 126976.2-e consists of curves linked by isogenies of degrees dividing 25.

Base change

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