Properties

Label 2.2.21.1-2268.1-m2
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 2268 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / 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(Polrev([-5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(90a-255\right){x}+726a-2030\)
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([1,1]),K([-255,90]),K([-2030,726])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([1,1]),Polrev([-255,90]),Polrev([-2030,726])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,0],K![1,1],K![-255,90],K![-2030,726]]);
 

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: \((40824a-163296)\) = \((-a+2)^{12}\cdot(2)^{3}\cdot(a+3)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 11666192832 \) = \(3^{12}\cdot4^{3}\cdot7^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2694303}{196} a - \frac{15380361}{392} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.2383689915264508585647723221546229761 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 0.27023426866457563274111588172504206851 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(1\) \(II^{*}\) Additive \(1\) \(4\) \(12\) \(0\)
\((2)\) \(4\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a+3)\) \(7\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 2268.1-m consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is not a \(\Q\)-curve.

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