Properties

Label 2.0.8.1-26136.4-d2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 26136 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

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

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

Weierstrass equation

\({y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a-21\right){x}-25a-14\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-21,6]),K([-14,-25])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-21,6]),Polrev([-14,-25])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-21,6],K![-14,-25]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((30a+156)\) = \((a)^{3}\cdot(-a-1)^{2}\cdot(a-1)\cdot(a+3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 26136 \) = \(2^{3}\cdot3^{2}\cdot3\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1057920a+858576)\) = \((a)^{8}\cdot(-a-1)^{7}\cdot(a-1)\cdot(a+3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2975542200576 \) = \(2^{8}\cdot3^{7}\cdot3\cdot11^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2048}{3} \)
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 $\left(-a + 9 : 6 a - 21 : 1\right)$
Height \(1.5331660019292994760082795683750635261\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.5331660019292994760082795683750635261 \)
Period: \( 1.2656653749340822343309650639889952293 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 5.4884924718494598226298782681895571118 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(3\) \(8\) \(0\)
\((-a-1)\) \(3\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((a-1)\) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((a+3)\) \(11\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 26136.4-d consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve.

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