Properties

Label 2.0.4.1-16200.2-d1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 16200 \)
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

\({y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i-9\right){x}-9i\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,1]),K([-9,-1]),K([0,-9])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([1,1]),Polrev([-9,-1]),Polrev([0,-9])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,1],K![-9,-1],K![0,-9]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((90i+90)\) = \((i+1)^{3}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16200 \) = \(2^{3}\cdot5\cdot5\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((43740)\) = \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1913187600 \) = \(2^{4}\cdot5\cdot5\cdot9^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{21296}{15} \)
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 $\left(-3 i : -3 i - 6 : 1\right)$
Height \(1.4975362857218438153786031768718604524\)
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: $\left(2 i : 3 i - 4 : 1\right)$
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: \( 1.4975362857218438153786031768718604524 \)
Period: \( 2.3210579238296434060638927348241762716 \)
Tamagawa product: \( 8 \)  =  \(2\cdot1\cdot1\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.4758684621970984666435975087353492155 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(III\) Additive \(1\) \(3\) \(4\) \(0\)
\((-i-2)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((3)\) \(9\) \(4\) \(I_{1}^{*}\) Additive \(1\) \(2\) \(7\) \(1\)

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 and 4.
Its isogeny class 16200.2-d consists of curves linked by isogenies of degrees dividing 4.

Base change

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

Base field Curve
\(\Q\) 360.e4
\(\Q\) 720.f4