Properties

Label 2.0.8.1-22050.2-b2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 22050 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 12 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-2}) \)

Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
gp: K = nfinit(Polrev([2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+1922{x}+20756\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([1922,0]),K([20756,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([1922,0]),Polrev([20756,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![1922,0],K![20756,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((105a)\) = \((a)\cdot(-a-1)\cdot(a-1)\cdot(5)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 22050 \) = \(2\cdot3\cdot3\cdot25\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-637994920500)\) = \((a)^{4}\cdot(-a-1)^{12}\cdot(a-1)^{12}\cdot(5)^{3}\cdot(7)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 407037518583801320250000 \) = \(2^{4}\cdot3^{12}\cdot3^{12}\cdot25^{3}\cdot49^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{785793873833639}{637994920500} \)
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 $\left(-35 a - 40 : -140 a + 492 : 1\right)$
Height \(1.0298047754258253638851671909413058041\)
Torsion structure: \(\Z/12\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-5 : 107 : 1\right)$
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: \( 1.0298047754258253638851671909413058041 \)
Period: \( 0.14868612716962054396510434565221573960 \)
Tamagawa product: \( 3456 \)  =  \(2\cdot( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot3\cdot2^{2}\)
Torsion order: \(12\)
Leading coefficient: \( 5.1969865216196224226681647932447956894 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-a-1)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((a-1)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((5)\) \(25\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((7)\) \(49\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 22050.2-b consists of curves linked by isogenies of degrees dividing 12.

Base change

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

Base field Curve
\(\Q\) 210.b8
\(\Q\) 6720.bi8