Properties

 Label 2.0.11.1-3249.2-a1 Base field $$\Q(\sqrt{-11})$$ Conductor $$(57)$$ Conductor norm $$3249$$ CM no Base change yes: 6897.f1,57.a1 Q-curve yes Torsion order $$1$$ Rank $$2$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-11})$$

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

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

gp: K = nfinit(a^2 - a + 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);

Weierstrass equation

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

gp: E = ellinit([0, -1, 1, -2, 2],K)

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(57)$$ = $$\left(-a\right) \cdot \left(a - 1\right) \cdot \left(19\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$3249$$ = $$3^{2} \cdot 361$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(171)$$ = $$\left(-a\right)^{2} \cdot \left(a - 1\right)^{2} \cdot \left(19\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$29241$$ = $$3^{4} \cdot 361$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1404928}{171}$$ 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: $$2$$ Generators $\left(\frac{2}{25} a + \frac{32}{25} : -\frac{8}{125} a - \frac{103}{125} : 1\right)$ $\left(2 : -2 : 1\right)$ Heights $$1.47070127199463$$ $$0.0375745927368237$$ 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: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.0520843387892159$$ Period: $$5.32864411591123$$ Tamagawa product: $$4$$  =  $$2\cdot2\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$2.67779611346913$$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(19\right)$$ $$361$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 3249.2-a consists of this curve only.

Base change

This curve is the base change of elliptic curves 6897.f1, 57.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.