# Properties

 Label 2.0.11.1-38025.5-b4 Base field $$\Q(\sqrt{-11})$$ Conductor $$(195)$$ Conductor norm $$38025$$ CM no Base change yes: 23595.p5,195.a5 Q-curve yes Torsion order $$8$$ Rank $$0$$

# 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: 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

$$y^2+xy=x^{3}-115x+392$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-115,0]),K([392,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-115,0])),Pol(Vecrev([392,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-115,0],K![392,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(195)$$ = $$\left(-a\right) \cdot \left(a - 1\right) \cdot \left(-a - 1\right) \cdot \left(a - 2\right) \cdot \left(13\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$38025$$ = $$3^{2} \cdot 5^{2} \cdot 169$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(27720225)$$ = $$\left(-a\right)^{8} \cdot \left(a - 1\right)^{8} \cdot \left(-a - 1\right)^{2} \cdot \left(a - 2\right)^{2} \cdot \left(13\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$768410874050625$$ = $$3^{16} \cdot 5^{4} \cdot 169^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{168288035761}{27720225}$$ 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: $$\Z/2\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-1 : -22 : 1\right)$ $\left(4 : -2 : 1\right)$ 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: $$0.741268334868174$$ Tamagawa product: $$512$$  =  $$2^{3}\cdot2^{3}\cdot2\cdot2\cdot2$$ Torsion order: $$8$$ Leading coefficient: $$3.57601299742437$$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$3$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(a - 1\right)$$ $$3$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(-a - 1\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(a - 2\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(13\right)$$ $$169$$ $$2$$ $$I_{2}$$ 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
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 38025.5-b consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 23595.p5, 195.a5, defined over $$\Q$$, so it is also a $$\Q$$-curve.