Properties

Label 2.2.8.1-686.2-d7
Base field \(\Q(\sqrt{2}) \)
Conductor norm \( 686 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-103060a-164599\right){x}-23909933a-32816619\)
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,0]),K([-164599,-103060]),K([-32816619,-23909933])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,0]),Polrev([-164599,-103060]),Polrev([-32816619,-23909933])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,0],K![-164599,-103060],K![-32816619,-23909933]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((7a-28)\) = \((a)\cdot(-2a+1)^{2}\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 686 \) = \(2\cdot7^{2}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((143248a+1026368)\) = \((a)^{9}\cdot(-2a+1)^{10}\cdot(2a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1012391292416 \) = \(2^{9}\cdot7^{10}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{218768831290078842857759125}{76832} a + \frac{9668307757341907702465000}{2401} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(199 a + \frac{1443}{4} : -\frac{199}{2} a - \frac{1447}{8} : 1\right)$
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: \( 0.66175315233369934976592010281121399718 \)
Tamagawa product: \( 36 \)  =  \(3^{2}\cdot2^{2}\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 2.1056856366902993881518569538570464891 \)
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\((j)_{-}\)
\((a)\) \(2\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((-2a+1)\) \(7\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)
\((2a+1)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class 686.2-d consists of curves linked by isogenies of degrees dividing 36.

Base change

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

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