Properties

Label 2.0.4.1-76050.5-a4
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 76050 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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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

\({y}^2+i{x}{y}={x}^{3}-{x}^{2}-483{x}+4293\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([-483,0]),K([4293,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([-483,0]),Polrev([4293,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![-483,0],K![4293,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((195i+195)\) = \((i+1)\cdot(-i-2)\cdot(2i+1)\cdot(3)\cdot(-3i-2)\cdot(2i+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 76050 \) = \(2\cdot5\cdot5\cdot9\cdot13\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2570490)\) = \((i+1)^{2}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}\cdot(-3i-2)^{4}\cdot(2i+3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6607418840100 \) = \(2^{2}\cdot5\cdot5\cdot9^{2}\cdot13^{4}\cdot13^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{12501706118329}{2570490} \)
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: \(2\)
Generators $\left(3 i + 12 : -24 i + 9 : 1\right)$ $\left(\frac{27}{2} : -\frac{27}{4} i - \frac{9}{4} : 1\right)$
Heights \(0.29384343632105164754526704138199961813\) \(1.5108357058376209331074823660557212963\)
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: $\left(\frac{53}{4} : -\frac{53}{8} i : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.44394915551986808557921340622342161019 \)
Period: \( 0.63504139978838126463628957015265435474 \)
Tamagawa product: \( 16 \)  =  \(2\cdot1\cdot1\cdot2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.5108174904993087683942688410439706517 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-i-2)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3)\) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-3i-2)\) \(13\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2i+3)\) \(13\) \(2\) \(I_{4}\) Non-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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 76050.5-a 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
\(\Q\) 390.a1
\(\Q\) 3120.q1