Properties

Label 2.2.21.1-2100.1-d6
Base field \(\Q(\sqrt{21}) \)
Conductor \((-20 a + 10)\)
Conductor norm \( 2100 \)
CM no
Base change yes: 210.d5,4410.f5
Q-curve yes
Torsion order \( 12 \)
Rank \( 1 \)

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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

\(y^2+xy=x^{3}-361x+2585\)
sage: E = EllipticCurve(K, [1, 0, 0, -361, 2585])
 
gp: E = ellinit([1, 0, 0, -361, 2585],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -361, 2585]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-20 a + 10)\) = \( \left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 2100 \) = \( 3 \cdot 4 \cdot 5^{2} \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((57153600)\) = \( \left(2\right)^{6} \cdot \left(-a + 2\right)^{12} \cdot \left(-a\right)^{2} \cdot \left(-a + 1\right)^{2} \cdot \left(a + 3\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3266533992960000 \) = \( 3^{12} \cdot 4^{6} \cdot 5^{4} \cdot 7^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5203798902289}{57153600} \)
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 $\left(-6 a - 1 : -9 a - 25 : 1\right)$
Height \(0.362952386387587\)
Torsion structure: \(\Z/2\Z\times\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(8 : 11 : 1\right)$ $\left(10 : -5 : 1\right)$
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: \( 0.362952386387587 \)
Period: \( 3.96179112472141 \)
Tamagawa product: \( 1152 \)  =  \(( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2^{2}\cdot( 2 \cdot 3 )\)
Torsion order: \(12\)
Leading coefficient: \(5.02055311704142\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\( \left(-a\right) \) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-a + 1\right) \) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a + 3\right) \) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2\right) \) \(4\) \(6\) \(I_{6}\) 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.1.1

Isogenies and isogeny class

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

Base change

This curve is the base change of elliptic curves 210.d5, 4410.f5, defined over \(\Q\), so it is also a \(\Q\)-curve.