Properties

 Label 2.2.12.1-1800.1-b1 Base field $$\Q(\sqrt{3})$$ Conductor $$(30a+30)$$ Conductor norm $$1800$$ CM no Base change yes: 720.i2,360.c2 Q-curve yes Torsion order $$2$$ Rank $$0$$

Related objects

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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(Pol(Vecrev([-3, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);

Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a-3\right){x}-36a-63$$
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([0,0]),K([-3,-3]),K([-63,-36])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-3,-3])),Pol(Vecrev([-63,-36]))], K);

magma: E := EllipticCurve([K![1,1],K![-1,1],K![0,0],K![-3,-3],K![-63,-36]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(30a+30)$$ = $$(a+1)^{3}\cdot(a)^{2}\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1800$$ = $$2^{3}\cdot3^{2}\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2160)$$ = $$(a+1)^{8}\cdot(a)^{6}\cdot(5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4665600$$ = $$2^{8}\cdot3^{6}\cdot25$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{108}{5}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(a + 3 : -2 a - 3 : 1\right)$ 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: $$4.11365742319854$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$2.37502122063758$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$2$$ $$I_1^{*}$$ Additive $$1$$ $$3$$ $$8$$ $$0$$
$$(a)$$ $$3$$ $$4$$ $$I_0^{*}$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$(5)$$ $$25$$ $$1$$ $$I_{1}$$ 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.
Its isogeny class 1800.1-b consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of elliptic curves 720.i2, 360.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.