Properties

 Label 2.2.21.1-2268.1-v1 Base field $$\Q(\sqrt{21})$$ Conductor $$(18a+54)$$ Conductor norm $$2268$$ CM no Base change yes: 378.e1,2646.f1 Q-curve yes Torsion order $$1$$ Rank $$1$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{21})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 5$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-5, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);

Weierstrass equation

$${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(5310a-14868\right){x}+326160a-910530$$
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-14868,5310]),K([-910530,326160])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-14868,5310])),Pol(Vecrev([-910530,326160]))], K);

magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-14868,5310],K![-910530,326160]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(18a+54)$$ = $$(-a+2)^{4}\cdot(2)\cdot(a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2268$$ = $$3^{4}\cdot4\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3402)$$ = $$(-a+2)^{10}\cdot(2)\cdot(a+3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$11573604$$ = $$3^{10}\cdot4\cdot7^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{545407363875}{14}$$ 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 $\left(-55 a + 165 : 650 a - 1860 : 1\right)$ Height $$3.81848645795421$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$3.81848645795421$$ Period: $$0.174839084422365$$ Tamagawa product: $$2$$  =  $$1\cdot1\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$5.24472391568351$$ Analytic order of Ш: $$9$$ (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$$(j)_{-}$$
$$(-a+2)$$ $$3$$ $$1$$ $$IV^{*}$$ Additive $$-1$$ $$4$$ $$10$$ $$0$$
$$(2)$$ $$4$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(a+3)$$ $$7$$ $$2$$ $$I_{2}$$ 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
$$3$$ 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 2268.1-v consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 378.e1, 2646.f1, defined over $$\Q$$, so it is also a $$\Q$$-curve.