# Properties

 Label 2.2.12.1-2304.1-x3 Base field $$\Q(\sqrt{3})$$ Conductor norm $$2304$$ CM no Base change no Q-curve no Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{3})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))

gp: K = nfinit(Polrev([-3, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);

## Weierstrass equation

$${y}^2={x}^{3}+a{x}^{2}+\left(95492a-165396\right){x}+21042496a-36446672$$
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([-165396,95492]),K([-36446672,21042496])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([-165396,95492]),Polrev([-36446672,21042496])], K);

magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![-165396,95492],K![-36446672,21042496]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(48)$$ = $$(a+1)^{8}\cdot(a)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2304$$ = $$2^{8}\cdot3^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-124416)$$ = $$(a+1)^{18}\cdot(a)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$15479341056$$ = $$2^{18}\cdot3^{10}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1122088}{9} a + \frac{1989808}{9}$$ 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: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(92 a - 160 : 0 : 1\right)$ $\left(-192 a + 332 : 0 : 1\right)$ 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: $$5.0217223685650400542834776225909293221$$ Tamagawa product: $$16$$  =  $$2^{2}\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$1.4496463806432954813140897658083754576$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$4$$ $$I_{6}^{*}$$ Additive $$-1$$ $$8$$ $$18$$ $$0$$
$$(a)$$ $$3$$ $$4$$ $$I_{4}^{*}$$ Additive $$-1$$ $$2$$ $$10$$ $$4$$

## 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 and 4.
Its isogeny class 2304.1-x consists of curves linked by isogenies of degrees dividing 8.

## Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.