Properties

Label 2.0.4.1-38025.5-a7
Base field \(\Q(\sqrt{-1}) \)
Conductor \((195)\)
Conductor norm \( 38025 \)
CM no
Base change yes: 195.a4,3120.k4
Q-curve yes
Torsion order \( 16 \)
Rank \( 0 \)

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

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

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

Weierstrass equation

\(y^2+ixy=x^{3}-520x+4225\)
sage: E = EllipticCurve(K, [i, 0, 0, -520, 4225])
 
gp: E = ellinit([i, 0, 0, -520, 4225],K)
 
magma: E := ChangeRing(EllipticCurve([i, 0, 0, -520, 4225]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((195)\) = \( \left(3\right) \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(2 i + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 38025 \) = \( 5^{2} \cdot 9 \cdot 13^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((1445900625)\) = \( \left(3\right)^{4} \cdot \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(2 i + 3\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2090628617375390625 \) = \( 5^{8} \cdot 9^{4} \cdot 13^{8} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{15551989015681}{1445900625} \)
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: \(\Z/4\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-65 : -455 i : 1\right)$ $\left(-15 i + 10 : -95 i - 45 : 1\right)$
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: \( 0.370634167434087 \)
Tamagawa product: \( 1024 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(16\)
Leading coefficient: \(1.48253666973635\)
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\((j)_{-}\)
\( \left(-i - 2\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2 i + 1\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-3 i - 2\right) \) \(13\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2 i + 3\right) \) \(13\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 38025.5-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of elliptic curves 195.a4, 3120.k4, defined over \(\Q\), so it is also a \(\Q\)-curve.