# Properties

 Label 2.0.4.1-13122.1-a2 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$13122$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{-1})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))

gp: K = nfinit(Polrev([1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);

## Weierstrass equation

$${y}^2+i{x}{y}={x}^{3}+{x}^{2}+39{x}+19$$
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([39,0]),K([19,0])])

gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([39,0]),Polrev([19,0])], K);

magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![39,0],K![19,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(81i+81)$$ = $$(i+1)\cdot(3)^{4}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$13122$$ = $$2\cdot9^{4}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3779136)$$ = $$(i+1)^{12}\cdot(3)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$14281868906496$$ = $$2^{12}\cdot9^{10}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{109503}{64}$$ 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: $$1$$ Generator $\left(-10 : -31 i : 1\right)$ Height $$0.10197829462292110456679008894485744001$$ 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: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.10197829462292110456679008894485744001$$ Period: $$1.1018611264310826782954406120478112478$$ Tamagawa product: $$6$$  =  $$2\cdot3$$ Torsion order: $$1$$ Leading coefficient: $$1.3483910230167920407341291519453420245$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$(3)$$ $$9$$ $$3$$ $$IV^{*}$$ Additive $$1$$ $$4$$ $$10$$ $$0$$

## 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
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 13122.1-a consists of curves linked by isogenies of degree 3.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 162.a2
$$\Q$$ 1296.c2