Properties

Label 2.0.8.1-41616.5-n1
Base field \(\Q(\sqrt{-2}) \)
Conductor \((204)\)
Conductor norm \( 41616 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4350a+369\right){x}-86669a-140167\)
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([369,4350]),K([-140167,-86669])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([369,4350])),Pol(Vecrev([-140167,-86669]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![369,4350],K![-140167,-86669]]);
 

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: \((-13042944a-119942208)\) = \((a)^{12}\cdot(-a-1)^{4}\cdot(a-1)^{12}\cdot(-2a+3)\cdot(2a+3)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14726370036289536 \) = \(2^{12}\cdot3^{4}\cdot3^{12}\cdot17\cdot17^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{372082589114986904}{2610969633} a - \frac{181318779827784209}{2610969633} \)
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: $\left(-23 a + \frac{63}{2} : -\frac{65}{4} a - 23 : 1\right)$
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: \( 0.221570995867971 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{2}\cdot2\cdot1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.25339482953999 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{4}^{*}\) Additive \(1\) \(4\) \(12\) \(0\)
\((-a-1)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a-1)\) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((-2a+3)\) \(17\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2a+3)\) \(17\) \(1\) \(I_{3}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 41616.5-n consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.