# Properties

 Label 2.2.12.1-1458.1-p3 Base field $$\Q(\sqrt{3})$$ Conductor norm $$1458$$ CM no Base change yes Q-curve yes Torsion order $$3$$ Rank $$1$$

# 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

$${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(5302a-9186\right){x}+563242a-975566$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([1,1]),K([-9186,5302]),K([-975566,563242])])

gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,1]),Polrev([-9186,5302]),Polrev([-975566,563242])], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![1,1],K![-9186,5302],K![-975566,563242]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(27a+27)$$ = $$(a+1)\cdot(a)^{6}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1458$$ = $$2\cdot3^{6}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-169869312)$$ = $$(a+1)^{42}\cdot(a)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$28855583159353344$$ = $$2^{42}\cdot3^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1159088625}{2097152}$$ 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 $\left(-51 a + 91 : -258 a + 448 : 1\right)$ Height $$1.0267244893349748108615965951594791037$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-75 a + 131 : 894 a - 1552 : 1\right)$ 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: $$1.0267244893349748108615965951594791037$$ Period: $$0.52085241842792041319755632697253449779$$ Tamagawa product: $$126$$  =  $$( 2 \cdot 3 \cdot 7 )\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$4.3225100752781365183402746128135951273$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$42$$ $$I_{42}$$ Split multiplicative $$-1$$ $$1$$ $$42$$ $$42$$
$$(a)$$ $$3$$ $$3$$ $$IV$$ Additive $$1$$ $$6$$ $$8$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.1
$$7$$ 7B.2.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 7 and 21.
Its isogeny class 1458.1-p consists of curves linked by isogenies of degrees dividing 21.

## 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.c2
$$\Q$$ 1296.f2