Properties

 Label 2.2.21.1-2100.1-n11 Base field $$\Q(\sqrt{21})$$ Conductor $$\left(-20a + 10\right)$$ Conductor norm $$2100$$ CM no Base change yes: 210.b1,4410.bi1 Q-curve yes Torsion order $$4$$ Rank $$1$$

Related objects

Show commands for: 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

$$y^2+xy+y=x^{3}-351233x-80149132$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-351233,0]),K([-80149132,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-351233,0])),Pol(Vecrev([-80149132,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-351233,0],K![-80149132,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$\left(-20a + 10\right)$$ = $$\left(-a + 2\right)\cdot\left(2\right)\cdot\left(-a\right)\cdot\left(-a + 1\right)\cdot\left(a + 3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2100$$ = $$3\cdot4\cdot5\cdot5\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 41160000 )$$ = $$\left(-a + 2\right)^{2}\cdot\left(2\right)^{6}\cdot\left(-a\right)^{4}\cdot\left(-a + 1\right)^{4}\cdot\left(a + 3\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1694145600000000$$ = $$3^{2}\cdot4^{6}\cdot5^{4}\cdot5^{4}\cdot7^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4791901410190533590281}{41160000}$$ 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 $\left(\frac{376831}{1156} a + \frac{279373}{1156} : -\frac{35422114}{4913} a - \frac{500904283}{39304} : 1\right)$ Height $$5.44682081111689$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{1369}{4} : \frac{1365}{8} : 1\right)$ $\left(-224 a + 283 : 112 a - 142 : 1\right)$ 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: $$5.44682081111689$$ Period: $$0.0384593330896625$$ Tamagawa product: $$1152$$  =  $$2\cdot( 2 \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )$$ Torsion order: $$4$$ Leading coefficient: $$6.58260328056218$$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$3$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(2\right)$$ $$4$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-a\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(-a + 1\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a + 3\right)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 2100.1-n consists of curves linked by isogenies of degrees dividing 24.

Base change

This curve is the base change of elliptic curves 210.b1, 4410.bi1, defined over $$\Q$$, so it is also a $$\Q$$-curve.