# Properties

 Label 2.0.3.1-57600.1-j3 Base field $$\Q(\sqrt{-3})$$ Conductor $$(240)$$ Conductor norm $$57600$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

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

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

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

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

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

## Weierstrass equation

$${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+1680a{x}-17400a+8700$$
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([0,1680]),K([8700,-17400])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1680])),Pol(Vecrev([8700,-17400]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![0,1680],K![8700,-17400]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(240)$$ = $$(-2a+1)^{2}\cdot(2)^{4}\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$57600$$ = $$3^{2}\cdot4^{4}\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(388800000000)$$ = $$(-2a+1)^{10}\cdot(2)^{12}\cdot(5)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$151165440000000000000000$$ = $$3^{10}\cdot4^{12}\cdot25^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4733169839}{3515625}$$ 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 $\left(-\frac{1530}{49} a + \frac{1924}{49} : -\frac{32722}{343} a + \frac{72232}{343} : 1\right)$ Height $$5.32932148184013$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(5 a + 5 : 0 : 1\right)$ 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: $$1$$ Regulator: $$5.32932148184013$$ Period: $$0.161347873200930$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$3.97159105467979$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$I_{4}^{*}$$ Additive $$-1$$ $$2$$ $$10$$ $$4$$
$$(2)$$ $$4$$ $$2$$ $$I_{4}^{*}$$ Additive $$-1$$ $$4$$ $$12$$ $$0$$
$$(5)$$ $$25$$ $$2$$ $$I_{8}$$ Non-split multiplicative $$1$$ $$1$$ $$8$$ $$8$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 57600.1-j consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.