# Properties

 Label 2.2.24.1-150.1-e8 Base field $$\Q(\sqrt{6})$$ Conductor norm $$150$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

# Learn more

Show commands: Magma / PariGP / 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(Polrev([-6, 0, 1]));

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

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}+\left(-1310a-1414\right){x}+25288a+58064$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-1414,-1310]),K([58064,25288])])

gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-1414,-1310]),Polrev([58064,25288])], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-1414,-1310],K![58064,25288]]);

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: $$(89116843750a-159057000000)$$ = $$(-a+2)^{3}\cdot(a+3)\cdot(-a-1)^{6}\cdot(-a+1)^{24}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-22351741790771484375000$$ = $$-2^{3}\cdot3\cdot5^{6}\cdot5^{24}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{26673883482189453500771}{715255737304687500} a + \frac{5545867307448315927528}{59604644775390625}$$ 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 $\left(\frac{361}{50} a + \frac{979539}{18050} : \frac{35384591}{3429500} a + \frac{30665863}{90250} : 1\right)$ Height $$7.5251185458064806048067757969054486758$$ 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: $\left(8 a + \frac{87}{4} : -4 a - \frac{91}{8} : 1\right)$ 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: $$7.5251185458064806048067757969054486758$$ Period: $$0.62419761847956101597909223019525289610$$ Tamagawa product: $$4$$  =  $$1\cdot1\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.9176079789302314243564354135619484477$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(a+3)$$ $$3$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(-a-1)$$ $$5$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(-a+1)$$ $$5$$ $$2$$ $$I_{24}$$ Non-split multiplicative $$1$$ $$1$$ $$24$$ $$24$$

## 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
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 150.1-e consists of curves linked by isogenies of degrees dividing 24.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.