Properties

Label 2.2.8.1-1458.1-k2
Base field \(\Q(\sqrt{2}) \)
Conductor \((27 a)\)
Conductor norm \( 1458 \)
CM no
Base change yes: 54.a1,1728.c1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

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

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

Weierstrass equation

\(y^2+xy=x^{3}-x^{2}-123x-667\)
sage: E = EllipticCurve(K, [1, -1, 0, -123, -667])
 
gp: E = ellinit([1, -1, 0, -123, -667],K)
 
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -123, -667]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((27 a)\) = \( \left(a\right) \cdot \left(3\right)^{3} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 1458 \) = \( 2 \cdot 9^{3} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((90699264)\) = \( \left(a\right)^{18} \cdot \left(3\right)^{11} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 8226356490141696 \) = \( 2^{18} \cdot 9^{11} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1167051}{512} \)
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 $\left(54 : -272 a - 27 : 1\right)$
Height \(2.54345832472117\)
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: \( 2.54345832472117 \)
Period: \( 0.492207235528415 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(1\)
Leading coefficient: \(1.77046610780906\)
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(I_{18}\) Non-split multiplicative \(1\) \(1\) \(18\) \(18\)
\( \left(3\right) \) \(9\) \(1\) \(II^*\) Additive \(-1\) \(3\) \(11\) \(0\)

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.1.2

Isogenies and isogeny class

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

Base change

This curve is the base change of elliptic curves 54.a1, 1728.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.