Properties

Label 2.0.8.1-41616.5-f1
Base field \(\Q(\sqrt{-2}) \)
Conductor \((204)\)
Conductor norm \( 41616 \)
CM no
Base change yes: 3264.a1,816.g1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

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(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-{x}^{2}-237{x}-1329\)
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-237,0]),K([-1329,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-237,0])),Pol(Vecrev([-1329,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![-237,0],K![-1329,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((204)\) = \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41616 \) = \(2^{4}\cdot3\cdot3\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-943296)\) = \((a)^{12}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)^{3}\cdot(2a+3)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 889807343616 \) = \(2^{12}\cdot3\cdot3\cdot17^{3}\cdot17^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{23100424192}{14739} \)
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 $\left(26 : -97 : 1\right)$
Height \(3.54275876238964\)
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: \( 3.54275876238964 \)
Period: \( 0.839708866198734 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\cdot1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 4.20712404792651 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(II^{*}\) Additive \(1\) \(4\) \(12\) \(0\)
\((-a-1)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((a-1)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-2a+3)\) \(17\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((2a+3)\) \(17\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 41616.5-f consists of curves linked by isogenies of degree 3.

Base change

This curve is the base change of 3264.a1, 816.g1, defined over \(\Q\), so it is also a \(\Q\)-curve.