Properties

Label 2.0.4.1-22050.2-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 22050 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-1}) \)

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

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

Weierstrass equation

\({y}^2+i{x}{y}+i{y}={x}^{3}-20352{x}+1443724\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([-20352,0]),K([1443724,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,1]),Polrev([-20352,0]),Polrev([1443724,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![-20352,0],K![1443724,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((105i+105)\) = \((i+1)\cdot(-i-2)\cdot(2i+1)\cdot(3)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 22050 \) = \(2\cdot5\cdot5\cdot9\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-358766164249920)\) = \((i+1)^{12}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{4}\cdot(7)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 128713160610600575476220006400 \) = \(2^{12}\cdot5\cdot5\cdot9^{4}\cdot49^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{932348627918877961}{358766164249920} \)
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 $\left(\frac{421}{2} : -\frac{423}{4} i + \frac{10179}{4} : 1\right)$
Height \(5.0891985548147912517953548275213578675\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-427 : -8019 i : 1\right)$
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: \( 5.0891985548147912517953548275213578675 \)
Period: \( 0.049562042389873514655034781884071913200 \)
Tamagawa product: \( 96 \)  =  \(2\cdot1\cdot1\cdot2^{2}\cdot( 2^{2} \cdot 3 )\)
Torsion order: \(4\)
Leading coefficient: \( 3.0267728940505645630281636143792505144 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((-i-2)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((3)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((7)\) \(49\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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.1.2

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.b3
\(\Q\) 1680.g3