# Properties

 Label 2.0.163.1-196.1-a5 Base field $$\Q(\sqrt{-163})$$ Conductor $$(14)$$ Conductor norm $$196$$ CM no Base change yes: 371966b2,14a6 Q-curve yes Torsion order $$6$$ Rank not available

# 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

$${y}^2+{x}{y}+{y}={x}^3-11{x}+12$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-11,0]),K([12,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-11,0])),Pol(Vecrev([12,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-11,0],K![12,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(14)$$ = $$(2)\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$196$$ = $$4\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(98)$$ = $$(2)\cdot(7)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$9604$$ = $$4\cdot49^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{128787625}{98}$$ 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: $$0 \le r \le 1$$ Torsion structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(0 : -4 : 1\right)$ 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: $$0 \le r \le 1$$ Regulator: not available Period: $$3.93937710837302$$ Tamagawa product: $$2$$  =  $$1\cdot2$$ Torsion order: $$6$$ Leading coefficient: $$6.22807824970597$$ Analytic order of Ш: not available

## 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(7)$$ $$49$$ $$2$$ $$I_{2}$$ 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
$$2$$ 2B3Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 196.1-a consists of curves linked by isogenies of degrees dividing 18.

## Base change

This curve is the base change of 371966b2, 14a6, defined over $$\Q$$, so it is also a $$\Q$$-curve.