# Properties

 Label 2.2.24.1-150.1-b7 Base field $$\Q(\sqrt{6})$$ Conductor norm $$150$$ CM no Base change yes Q-curve yes Torsion order $$8$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{6})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 6$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-6, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-371a-906\right){x}-5568a-13638$$
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([-906,-371]),K([-13638,-5568])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-906,-371])),Pol(Vecrev([-13638,-5568]))], K);

magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![-906,-371],K![-13638,-5568]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-5a)$$ = $$(-a+2)\cdot(a+3)\cdot(-a-1)\cdot(-a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$150$$ = $$2\cdot3\cdot5\cdot5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(72900)$$ = $$(-a+2)^{4}\cdot(a+3)^{12}\cdot(-a-1)^{2}\cdot(-a+1)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$5314410000$$ = $$2^{4}\cdot3^{12}\cdot5^{2}\cdot5^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{702595369}{72900}$$ 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: $$\Z/2\Z\oplus\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{7}{2} a - \frac{39}{4} : \frac{53}{8} a + \frac{123}{8} : 1\right)$ $\left(-5 a - 12 : -2 a - 6 : 1\right)$ 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: $$5.36748913421879$$ Tamagawa product: $$192$$  =  $$2^{2}\cdot( 2^{2} \cdot 3 )\cdot2\cdot2$$ Torsion order: $$8$$ Leading coefficient: $$3.28690239469227$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(a+3)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$(-a-1)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-a+1)$$ $$5$$ $$2$$ $$I_{2}$$ 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
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 150.1-b consists of curves linked by isogenies of degrees dividing 24.

## Base change

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

Base field Curve
$$\Q$$ 90.c6
$$\Q$$ 960.p6