# Properties

 Label 2.0.3.1-71148.2-e3 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$71148$$ CM no Base change yes Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([1, -1, 1]));

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

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2517a+2517\right){x}+48285$$
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([2517,-2517]),K([48285,0])])

gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([2517,-2517]),Polrev([48285,0])], K);

magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![2517,-2517],K![48285,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-308a+154)$$ = $$(-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(11)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$71148$$ = $$3\cdot4\cdot7\cdot7\cdot121$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(4612311396)$$ = $$(-2a+1)^{8}\cdot(2)^{2}\cdot(-3a+1)^{6}\cdot(3a-2)^{6}\cdot(11)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$21273416413671468816$$ = $$3^{8}\cdot4^{2}\cdot7^{6}\cdot7^{6}\cdot121^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1763535241378513}{4612311396}$$ 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: $\left(-30 a : 30 a - 15 : 1\right)$ $\left(-\frac{111}{4} a : \frac{111}{4} a - \frac{111}{8} : 1\right)$ 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: $$0.23239578798352641403107840948678370377$$ Tamagawa product: $$128$$  =  $$2^{3}\cdot2\cdot2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$2.1467803320131867467924918687634274992$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$(2)$$ $$4$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-3a+1)$$ $$7$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(3a-2)$$ $$7$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(11)$$ $$121$$ $$2$$ $$I_{2}$$ 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.
Its isogeny class 71148.2-e consists of curves linked by isogenies of degrees dividing 4.

## Base change

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

Base field Curve
$$\Q$$ 462.g3
$$\Q$$ 1386.a3