Properties

Label 2.0.8.1-41616.5-e3
Base field \(\Q(\sqrt{-2}) \)
Conductor \((204)\)
Conductor norm \( 41616 \)
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

\({y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-4360a+418\right){x}-97280a+146930\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([418,-4360]),K([146930,-97280])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([418,-4360])),Pol(Vecrev([146930,-97280]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![418,-4360],K![146930,-97280]]);
 

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: \((1788724321920a+1783310266368)\) = \((a)^{15}\cdot(-a-1)^{6}\cdot(a-1)^{24}\cdot(-2a+3)^{4}\cdot(2a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9579264905789834696884224 \) = \(2^{15}\cdot3^{6}\cdot3^{24}\cdot17^{4}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{410311536960329978125}{94355189265718404} a - \frac{136155893118005119000}{23588797316429601} \)
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 $\left(-\frac{8176}{361} a - \frac{16127}{722} : -\frac{2080851}{27436} a + \frac{1227905}{13718} : 1\right)$
Height \(6.29315391605464\)
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: $\left(-16 a - \frac{73}{2} : \frac{73}{4} a - 16 : 1\right)$
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: \( 6.29315391605464 \)
Period: \( 0.103966983730335 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.70116790397649 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{7}^{*}\) Additive \(1\) \(4\) \(15\) \(3\)
\((-a-1)\) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((a-1)\) \(3\) \(2\) \(I_{24}\) Non-split multiplicative \(1\) \(1\) \(24\) \(24\)
\((-2a+3)\) \(17\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2a+3)\) \(17\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.