# Properties

 Label 2.2.24.1-600.1-o1 Base field $$\Q(\sqrt{6})$$ Conductor norm $$600$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

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

$${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(2450a-6801\right){x}-109970a+277443$$
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-6801,2450]),K([277443,-109970])])

gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-6801,2450]),Polrev([277443,-109970])], K);

magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-6801,2450],K![277443,-109970]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-10a)$$ = $$(-a+2)^{3}\cdot(a+3)\cdot(-a-1)\cdot(-a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$600$$ = $$2^{3}\cdot3\cdot5\cdot5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-160a)$$ = $$(-a+2)^{11}\cdot(a+3)\cdot(-a-1)\cdot(-a+1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-153600$$ = $$-2^{11}\cdot3\cdot5\cdot5$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1705382858958144777649}{15} a + 278487854702414976744$$ 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 $\left(-9 a + \frac{382}{9} : -\frac{263}{9} a + \frac{233}{27} : 1\right)$ Height $$1.5209290178094052976036537061296142087$$ 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: $\left(-20 a + \frac{31}{2} : -\frac{33}{4} a + 60 : 1\right)$ 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: $$1.5209290178094052976036537061296142087$$ Period: $$5.8220381230590254037504628952170238605$$ Tamagawa product: $$1$$  =  $$1\cdot1\cdot1\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$3.6150005323524591425084886467933760700$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$3$$ $$11$$ $$0$$
$$(a+3)$$ $$3$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(-a-1)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(-a+1)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 600.1-o consists of curves linked by isogenies of degrees dividing 16.

## Base change

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

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