Properties

Label 2.0.8.1-3249.5-a1
Base field \(\Q(\sqrt{-2}) \)
Conductor \((57)\)
Conductor norm \( 3249 \)
CM no
Base change yes: 3648.r1,57.a1
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((57)\) = \( \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(-3 a + 1\right) \cdot \left(3 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 3249 \) = \( 3^{2} \cdot 19^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((171)\) = \( \left(-a - 1\right)^{2} \cdot \left(a - 1\right)^{2} \cdot \left(-3 a + 1\right) \cdot \left(3 a + 1\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29241 \) = \( 3^{4} \cdot 19^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1404928}{171} \)
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 $\left(2 : 1 : 1\right)$
Height \(0.0375745927368237\)
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: \( 0.0375745927368237 \)
Period: \( 5.32864411591123 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \(1.13262459261872\)
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a - 1\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-3 a + 1\right) \) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(3 a + 1\right) \) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 3249.5-a consists of this curve only.

Base change

This curve is the base change of elliptic curves 3648.r1, 57.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.