Properties

Label 2.2.21.1-21.1-b5
Base field \(\Q(\sqrt{21}) \)
Conductor \((-2 a + 1)\)
Conductor norm \( 21 \)
CM no
Base change yes: 21.a2,441.f2
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

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: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)
 
gp: K = nfinit(a^2 - a - 5);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\(y^2+xy=x^{3}-49x-136\)
sage: E = EllipticCurve(K, [1, 0, 0, -49, -136])
 
gp: E = ellinit([1, 0, 0, -49, -136],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -49, -136]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2 a + 1)\) = \( \left(-a + 2\right) \cdot \left(a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 21 \) = \( 3 \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((21609)\) = \( \left(-a + 2\right)^{4} \cdot \left(a + 3\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 466948881 \) = \( 3^{4} \cdot 7^{8} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{13027640977}{21609} \)
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 $\left(-4 a + 7 : -8 a + 24 : 1\right)$
Height \(1.31925441156708\)
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: $\left(-\frac{17}{4} : \frac{17}{8} : 1\right)$ $\left(-4 : 2 : 1\right)$
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: \( 1.31925441156708 \)
Period: \( 3.25608174366118 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \(0.937376813977413\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(a + 3\right) \) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 21.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 21.a2, 441.f2, defined over \(\Q\), so it is also a \(\Q\)-curve.