Properties

Label 2.0.4.1-5625.3-b8
Base field \(\Q(\sqrt{-1}) \)
Conductor \((75)\)
Conductor norm \( 5625 \)
CM no
Base change yes: 75.b2,1200.e2
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

\({y}^2+{x}{y}+{y}={x}^{3}-3376{x}-75727\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-3376,0]),K([-75727,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-3376,0])),Pol(Vecrev([-75727,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-3376,0],K![-75727,0]]);
 

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: \((2562890625)\) = \((-i-2)^{8}\cdot(2i+1)^{8}\cdot(3)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6568408355712890625 \) = \(5^{8}\cdot5^{8}\cdot9^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{272223782641}{164025} \)
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: $\left(-\frac{137}{4} : \frac{133}{8} : 1\right)$ $\left(-33 : 16 : 1\right)$
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: \( 128 \)  =  \(2^{2}\cdot2^{2}\cdot2^{3}\)
Torsion order: \(4\)
Leading coefficient: \( 1.78856137009402 \)
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\((j)_{-}\)
\((-i-2)\) \(5\) \(4\) \(I_2^{*}\) Additive \(1\) \(2\) \(8\) \(2\)
\((2i+1)\) \(5\) \(4\) \(I_2^{*}\) Additive \(1\) \(2\) \(8\) \(2\)
\((3)\) \(9\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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 75.b2, 1200.e2, defined over \(\Q\), so it is also a \(\Q\)-curve.