Properties

Label 2.0.8.1-46818.8-k3
Base field \(\Q(\sqrt{-2}) \)
Conductor \((153a)\)
Conductor norm \( 46818 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
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

\({y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-38745a-8316\right){x}-3718953a+3437478\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-8316,-38745]),K([3437478,-3718953])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-8316,-38745])),Pol(Vecrev([3437478,-3718953]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-8316,-38745],K![3437478,-3718953]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((153a)\) = \((a)\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 46818 \) = \(2\cdot3^{2}\cdot3^{2}\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4136899013169237a+2156459775493680)\) = \((a)\cdot(-a-1)^{7}\cdot(a-1)^{7}\cdot(-2a+3)^{4}\cdot(2a+3)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 38878185653643466599797326266738 \) = \(2\cdot3^{7}\cdot3^{7}\cdot17^{4}\cdot17^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{61437923106397764191713}{291967151254001210886} a - \frac{146204680417750243067160}{48661191875666868481} \)
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 $\left(-\frac{56677}{33282} a - \frac{53838098399}{12014802} : \frac{12490350017994377}{58896559404} a + \frac{1859277270754}{774954729} : 1\right)$
Height \(13.6500309790533\)
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: $\left(-36 a - \frac{573}{4} : 18 a + \frac{573}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 13.6500309790533 \)
Period: \( 0.0306256912019913 \)
Tamagawa product: \( 32 \)  =  \(1\cdot2^{2}\cdot2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.72960118369238 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-a-1)\) \(3\) \(4\) \(I_1^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((a-1)\) \(3\) \(2\) \(I_1^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((-2a+3)\) \(17\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2a+3)\) \(17\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)

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, 4, 8 and 16.
Its isogeny class 46818.8-k consists of curves linked by isogenies of degrees dividing 16.

Base change

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