Properties

Label 2.0.4.1-60025.2-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor \((245)\)
Conductor norm \( 60025 \)
CM no
Base change yes: 3920.bj1,245.a1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+i{y}={x}^{3}-7{x}-12\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([-7,0]),K([-12,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-7,0])),Pol(Vecrev([-12,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![-7,0],K![-12,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((245)\) = \((-i-2)\cdot(2i+1)\cdot(7)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 60025 \) = \(5\cdot5\cdot49^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-42875)\) = \((-i-2)^{3}\cdot(2i+1)^{3}\cdot(7)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1838265625 \) = \(5^{3}\cdot5^{3}\cdot49^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{110592}{125} \)
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 $\left(-7 : 17 i : 1\right)$
Height \(0.0322368866282492\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0322368866282492 \)
Period: \( 2.30140922416324 \)
Tamagawa product: \( 18 \)  =  \(3\cdot3\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 2.67084965680406 \)
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\((j)_{-}\)
\((-i-2)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((2i+1)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((7)\) \(49\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3Nn

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 60025.2-b consists of this curve only.

Base change

This curve is the base change of 3920.bj1, 245.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.