# Properties

 Label 2.0.3.1-7203.3-b2 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$7203$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# 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

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

gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([1,0]),Polrev([2,-2]),Polrev([-1,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,1],K![1,0],K![2,-2],K![-1,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-98a+49)$$ = $$(-2a+1)\cdot(-3a+1)^{2}\cdot(3a-2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$7203$$ = $$3\cdot7^{2}\cdot7^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-147)$$ = $$(-2a+1)^{2}\cdot(-3a+1)^{2}\cdot(3a-2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$21609$$ = $$3^{2}\cdot7^{2}\cdot7^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{28672}{3}$$ 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 $\left(a : -a : 1\right)$ Height $$0.076270034394571012613674763317560287258$$ 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: $$0.076270034394571012613674763317560287258$$ Period: $$5.4204293752418674265143998805448040181$$ Tamagawa product: $$2$$  =  $$2\cdot1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$1.9094882578568942878120978565977180232$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(-3a+1)$$ $$7$$ $$1$$ $$II$$ Additive $$-1$$ $$2$$ $$2$$ $$0$$
$$(3a-2)$$ $$7$$ $$1$$ $$II$$ Additive $$-1$$ $$2$$ $$2$$ $$0$$

## 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
$$13$$ 13B.4.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 13.
Its isogeny class 7203.3-b consists of curves linked by isogenies of degree 13.

## Base change

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

Base field Curve
$$\Q$$ 147.b2
$$\Q$$ 441.a2