# Properties

 Label 2.0.163.1-121.1-a2 Base field $$\Q(\sqrt{-163})$$ Conductor $$(11)$$ Conductor norm $$121$$ CM no Base change yes: 11a1,292259e2 Q-curve yes Torsion order $$5$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-163})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 41$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, -1, 1]))

gp: K = nfinit(Pol(Vecrev([41, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -1, 1]);

## Weierstrass equation

$${y}^2+{y}={x}^3-{x}^2-10{x}-20$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-10,0]),K([-20,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-10,0])),Pol(Vecrev([-20,0]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-10,0],K![-20,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(11)$$ = $$(11)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$121$$ = $$121$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(161051)$$ = $$(11)^{5}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$25937424601$$ = $$121^{5}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{122023936}{161051}$$ 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: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(5 : -6 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$1.85154362345596$$ Tamagawa product: $$5$$ Torsion order: $$5$$ Leading coefficient: $$0.232038542668872$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$(11)$$ $$121$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 121.1-a consists of curves linked by isogenies of degrees dividing 25.

## Base change

This curve is the base change of 11a1, 292259e2, defined over $$\Q$$, so it is also a $$\Q$$-curve.