Properties

Label 2.0.8.1-2601.5-e3
Base field \(\Q(\sqrt{-2}) \)
Conductor \((51)\)
Conductor norm \( 2601 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-7a-158\right){x}+2a-1661\)
sage: E = EllipticCurve([K([1,1]),K([1,0]),K([0,0]),K([-158,-7]),K([-1661,2])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-158,-7])),Pol(Vecrev([-1661,2]))], K);
 
magma: E := EllipticCurve([K![1,1],K![1,0],K![0,0],K![-158,-7],K![-1661,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((51)\) = \((-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2601 \) = \(3\cdot3\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((76980012a-889055103)\) = \((-a-1)^{3}\cdot(a-1)\cdot(-2a+3)^{12}\cdot(2a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 802270820665380897 \) = \(3^{3}\cdot3\cdot17^{12}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1695948356939509768}{15730800405203547} a - \frac{7330062339265374821}{15730800405203547} \)
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: $\left(-\frac{9}{2} a - \frac{35}{4} : \frac{53}{8} a - \frac{1}{8} : 1\right)$
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: \( 0.443141991735942 \)
Tamagawa product: \( 6 \)  =  \(3\cdot1\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.88009224430998 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-a-1)\) \(3\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a-1)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-2a+3)\) \(17\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((2a+3)\) \(17\) \(1\) \(I_{1}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 2601.5-e 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.