Properties

Label 2.2.21.1-2268.1-l1
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 2268 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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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

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

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: \((-252)\) = \((-a+2)^{4}\cdot(2)^{2}\cdot(a+3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 63504 \) = \(3^{4}\cdot4^{2}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{3}{28} \)
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 $\left(1 : 1 : 1\right)$
Height \(0.20567776950624741381306022721218811076\)
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: \( 0.20567776950624741381306022721218811076 \)
Period: \( 11.175796821568442217169553997899100314 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 4.0127877695425294650770151235549797653 \)
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 \(1\) \(4\) \(4\) \(0\)
\((2)\) \(4\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a+3)\) \(7\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 2268.1-l consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve.

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