Properties

Label 2.2.21.1-21.1-b1
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 21 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
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

\({y}^2+{x}{y}={x}^{3}-34{x}-217\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-34,0]),K([-217,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-34,0]),Polrev([-217,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-34,0],K![-217,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+1)\) = \((-a+2)\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 21 \) = \(3\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-17294403)\) = \((-a+2)^{2}\cdot(a+3)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 299096375126409 \) = \(3^{2}\cdot7^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4354703137}{17294403} \)
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 $\left(14 : -20 a + 3 : 1\right)$
Height \(2.6385088231341581748855033900597912662\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{31}{4} : -\frac{31}{8} : 1\right)$
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: \( 2.6385088231341581748855033900597912662 \)
Period: \( 0.81402043591529410859009812409796429516 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 0.93737681397741291827004797814744196432 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a+3)\) \(7\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

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

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 21.a4
\(\Q\) 441.f4