Properties

Label 2.0.4.1-1458.1-a2
Base field \(\Q(\sqrt{-1}) \)
Conductor \((27 i + 27)\)
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(Pol(Vecrev([1, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\(y^2+xy+\left(i+1\right)y=x^{3}-x^{2}+\left(13i-2\right)x+13i+8\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,1]),K([-2,13]),K([8,13])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([-2,13])),Pol(Vecrev([8,13]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,1],K![-2,13],K![8,13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((27 i + 27)\) = \( \left(i + 1\right) \cdot \left(3\right)^{3} \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1458 \) = \( 2 \cdot 9^{3} \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((39366 i + 39366)\) = \( \left(i + 1\right)^{3} \cdot \left(3\right)^{9} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3099363912 \) = \( 2^{3} \cdot 9^{9} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{18333}{4} i + \frac{8019}{4} \)
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: $\left(3 i - 2 : -2 i - 4 : 1\right)$
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: \( 2.09827028885310 \)
Tamagawa product: \( 3 \)  =  \(1\cdot3\)
Torsion order: \(3\)
Leading coefficient: \(0.699423429617700\)
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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(3\right) \) \(9\) \(3\) \(IV^*\) Additive \(-1\) \(3\) \(9\) \(0\)

Galois Representations

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

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

Isogenies and isogeny class

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

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.