Properties

Label 2.0.8.1-41616.5-e2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 41616 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(34240a+89158\right){x}+6490880a-8469742\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([89158,34240]),K([-8469742,6490880])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([89158,34240]),Polrev([-8469742,6490880])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![89158,34240],K![-8469742,6490880]]);
 

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: \((420792346874880a+18302538350592)\) = \((a)^{21}\cdot(-a-1)^{8}\cdot(a-1)^{2}\cdot(-2a+3)^{3}\cdot(2a+3)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 354467381287013555132656779264 \) = \(2^{21}\cdot3^{8}\cdot3^{2}\cdot17^{3}\cdot17^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{795638018697416056308875}{122322703950862781472} a - \frac{740752568391229768255000}{3822584498464461921} \)
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 $\left(-\frac{9004547328}{137428729} a + \frac{2298087951135}{26936030884} : -\frac{503890782160613525}{631542180106264} a - \frac{8182058300901473015}{4420795260743848} : 1\right)$
Height \(18.879461748163909778440107247889587313\)
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: $\left(-128 a + \frac{71}{2} : -\frac{71}{4} a - 128 : 1\right)$
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: \( 18.879461748163909778440107247889587313 \)
Period: \( 0.034655661243444931366936081676238121903 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.7011679039764952941272628899277629300 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{13}^{*}\) Additive \(1\) \(4\) \(21\) \(9\)
\((-a-1)\) \(3\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((a-1)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+3)\) \(17\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((2a+3)\) \(17\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 41616.5-e consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.