Properties

Label 2.0.7.1-26896.1-b1
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 26896 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{-7}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(24a-32\right){x}+68a-65\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,1]),K([-32,24]),K([-65,68])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,1]),Polrev([-32,24]),Polrev([-65,68])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,1],K![-32,24],K![-65,68]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-123a+82)\) = \((a)^{4}\cdot(41)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 26896 \) = \(2^{4}\cdot1681\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-11767a-51414)\) = \((a)^{21}\cdot(41)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3525312512 \) = \(2^{21}\cdot1681\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{115013825}{20992} a + \frac{1953156251}{20992} \)
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: \(2\)
Generators $\left(-a - 2 : a + 7 : 1\right)$ $\left(\frac{3}{4} a - \frac{13}{4} : -\frac{15}{8} a - \frac{3}{8} : 1\right)$
Heights \(0.13840443814756540840193902085969334720\) \(1.7776155725583737233478093837779976766\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.24600210923519483047258896071057181594 \)
Period: \( 1.7143672719315384479105298671299770190 \)
Tamagawa product: \( 4 \)  =  \(2^{2}\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 5.1008629648324423576643140937176702865 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{13}^{*}\) Additive \(-1\) \(4\) \(21\) \(9\)
\((41)\) \(1681\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 26896.1-b consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

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