# Properties

 Label 2.0.3.1-14700.2-h6 Base field $$\Q(\sqrt{-3})$$ Conductor $$(-140 a + 70)$$ Conductor norm $$14700$$ CM no Base change yes: 630.f5,210.d5 Q-curve yes Torsion order $$12$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / 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(Pol(Vecrev([1, -1, 1])));

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

## Weierstrass equation

$$y^2+axy=x^{3}+361ax+2585$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([0,361]),K([2585,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,361])),Pol(Vecrev([2585,0]))], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![0,361],K![2585,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-140 a + 70)$$ = $$\left(2\right) \cdot \left(-2 a + 1\right) \cdot \left(5\right) \cdot \left(-3 a + 1\right) \cdot \left(3 a - 2\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$14700$$ = $$3 \cdot 4 \cdot 7^{2} \cdot 25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(57153600)$$ = $$\left(2\right)^{6} \cdot \left(-2 a + 1\right)^{12} \cdot \left(5\right)^{2} \cdot \left(-3 a + 1\right)^{2} \cdot \left(3 a - 2\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$3266533992960000$$ = $$3^{12} \cdot 4^{6} \cdot 7^{4} \cdot 25^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{5203798902289}{57153600}$$ 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: $$\Z/2\Z\times\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(38 a - 38 : 229 : 1\right)$ $\left(10 a - 10 : 5 : 1\right)$ 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: $$0.545369050370004$$ Tamagawa product: $$576$$  =  $$( 2^{2} \cdot 3 )\cdot2\cdot2\cdot( 2 \cdot 3 )\cdot2$$ Torsion order: $$12$$ Leading coefficient: $$2.51895174431050$$ 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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$\left(-3 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(3 a - 2\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(2\right)$$ $$4$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(5\right)$$ $$25$$ $$2$$ $$I_{2}$$ 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.1.1[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 14700.2-h consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base change of elliptic curves 630.f5, 210.d5, defined over $$\Q$$, so it is also a $$\Q$$-curve.