Properties

Label 2.0.8.1-392.1-b2
Base field \(\Q(\sqrt{-2}) \)
Conductor \((14 a)\)
Conductor norm \( 392 \)
CM no
Base change yes: 56.b1,448.h1
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: 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

\(y^2+axy+ay=x^{3}+x^{2}-9x+11\)
sage: E = EllipticCurve(K, [a, 1, a, -9, 11])
 
gp: E = ellinit([a, 1, a, -9, 11],K)
 
magma: E := ChangeRing(EllipticCurve([a, 1, a, -9, 11]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((14 a)\) = \( \left(a\right)^{3} \cdot \left(7\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 392 \) = \( 2^{3} \cdot 49 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((1568)\) = \( \left(a\right)^{10} \cdot \left(7\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2458624 \) = \( 2^{10} \cdot 49^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3543122}{49} \)
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: $\left(\frac{3}{2} : -\frac{5}{4} a : 1\right)$
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: \( 3.19786964344142 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \(2.26123531022804\)
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(III^*\) Additive \(1\) \(3\) \(10\) \(0\)
\( \left(7\right) \) \(49\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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.
Its isogeny class 392.1-b consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of elliptic curves 56.b1, 448.h1, defined over \(\Q\), so it is also a \(\Q\)-curve.