# Properties

 Label 2.0.11.1-11.1-a1 Base field $$\Q(\sqrt{-11})$$ Conductor $$(-2a+1)$$ Conductor norm $$11$$ CM no Base change yes: 11.a1,121.d1 Q-curve yes Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))

gp: K = nfinit(Pol(Vecrev([3, -1, 1])));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-7820,0]),K([-263580,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-7820,0])),Pol(Vecrev([-263580,0]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-7820,0],K![-263580,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2a+1)$$ = $$(-2a+1)$$ sage: E.conductor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor norm: $$11$$ = $$11$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E))  magma: Norm(Conductor(E)); Discriminant: $$(-11)$$ = $$(-2a+1)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$121$$ = $$11^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{52893159101157376}{11}$$ 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: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.370308724691192$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.446609125962073$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$11$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cn
$$5$$ 5B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5 and 25.
Its isogeny class 11.1-a consists of curves linked by isogenies of degrees dividing 25.

## Base change

This curve is the base change of 11.a1, 121.d1, defined over $$\Q$$, so it is also a $$\Q$$-curve.