Properties

 Label 2.0.3.1-130816.2-a2 Base field $$\Q(\sqrt{-3})$$ Conductor $$(400a-96)$$ Conductor norm $$130816$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

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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

$${y}^2={x}^{3}-a{x}^{2}+\left(2a+21\right){x}+47a-54$$
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([21,2]),K([-54,47])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([21,2])),Pol(Vecrev([-54,47]))], K);

magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![21,2],K![-54,47]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(400a-96)$$ = $$(2)^{4}\cdot(-3a+1)\cdot(9a-8)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$130816$$ = $$4^{4}\cdot7\cdot73$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(1130496a-606208)$$ = $$(2)^{14}\cdot(-3a+1)^{2}\cdot(9a-8)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$960193626112$$ = $$4^{14}\cdot7^{2}\cdot73$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{20198991}{14308} a - \frac{3200392}{3577}$$ 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 $\left(-2 a + 1 : -6 a + 2 : 1\right)$ Height $$0.499940384488220$$ 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: $\left(-4 a + 1 : 0 : 1\right)$ 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: $$0.499940384488220$$ Period: $$1.36258781394617$$ Tamagawa product: $$8$$  =  $$2^{2}\cdot2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$3.14638657307919$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$4$$ $$I_{6}^{*}$$ Additive $$-1$$ $$4$$ $$14$$ $$2$$
$$(-3a+1)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(9a-8)$$ $$73$$ $$1$$ $$I_{1}$$ 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 130816.2-a consists of curves linked by isogenies of degree 2.

Base change

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