Properties

Label 2.0.4.1-1458.1-f1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 1458 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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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

\({y}^2+i{x}{y}+{y}={x}^{3}+{x}^{2}+\left(31i+15\right){x}-2i-75\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([1,0]),K([15,31]),K([-75,-2])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([1,0]),Polrev([15,31]),Polrev([-75,-2])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![1,0],K![15,31],K![-75,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((27i+27)\) = \((i+1)\cdot(3)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1458 \) = \(2\cdot9^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3888i-3888)\) = \((i+1)^{9}\cdot(3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 30233088 \) = \(2^{9}\cdot9^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{23376651}{32} i - \frac{13799643}{32} \)
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: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-3 i + 2 : -3 : 1\right)$
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: \( 2.0982702888530991750335404821637048708 \)
Tamagawa product: \( 9 \)  =  \(3^{2}\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 2.0982702888530991750335404821637048708 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((3)\) \(9\) \(1\) \(IV\) Additive \(1\) \(3\) \(5\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 1458.1-f consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.