Properties

Label 2.0.4.1-5625.3-b5
Base field \(\Q(\sqrt{-1}) \)
Conductor \((75)\)
Conductor norm \( 5625 \)
CM no
Base change yes: 1200.e8,75.b8
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / 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(Pol(Vecrev([1, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{x}{y}+i{y}={x}^{3}+875{x}+5227\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([875,0]),K([5227,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([875,0])),Pol(Vecrev([5227,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![875,0],K![5227,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((75)\) = \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5625 \) = \(5^{2}\cdot5^{2}\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-54931640625)\) = \((-i-2)^{14}\cdot(2i+1)^{14}\cdot(3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3017485141754150390625 \) = \(5^{14}\cdot5^{14}\cdot9^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4733169839}{3515625} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{23}{4} : \frac{19}{8} i : 1\right)$ $\left(-30 i + 3 : -2 i - 15 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.223570171261753 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.78856137009402 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-i-2)\) \(5\) \(4\) \(I_8^{*}\) Additive \(1\) \(2\) \(14\) \(8\)
\((2i+1)\) \(5\) \(4\) \(I_8^{*}\) Additive \(1\) \(2\) \(14\) \(8\)
\((3)\) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 5625.3-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of 1200.e8, 75.b8, defined over \(\Q\), so it is also a \(\Q\)-curve.