Properties

Label 6.6.1868969.1-26.1-a3
Base field 6.6.1868969.1
Conductor norm \( 26 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 6.6.1868969.1

Generator \(a\), with minimal polynomial \( x^{6} - 6 x^{4} - x^{3} + 8 x^{2} + x - 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 1, 8, -1, -6, 0, 1]))
 
gp: K = nfinit(Polrev([-2, 1, 8, -1, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 1, 8, -1, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}+a^{4}-5a^{3}-6a^{2}+3a+3\right){x}{y}+\left(a^{4}-5a^{2}+4\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}^{2}+\left(-7a^{5}-6a^{4}+39a^{3}+38a^{2}-34a-31\right){x}+5a^{5}-29a^{3}-9a^{2}+34a+21\)
sage: E = EllipticCurve([K([3,3,-6,-5,1,1]),K([3,3,-5,-1,1,0]),K([4,0,-5,0,1,0]),K([-31,-34,38,39,-6,-7]),K([21,34,-9,-29,0,5])])
 
gp: E = ellinit([Polrev([3,3,-6,-5,1,1]),Polrev([3,3,-5,-1,1,0]),Polrev([4,0,-5,0,1,0]),Polrev([-31,-34,38,39,-6,-7]),Polrev([21,34,-9,-29,0,5])], K);
 
magma: E := EllipticCurve([K![3,3,-6,-5,1,1],K![3,3,-5,-1,1,0],K![4,0,-5,0,1,0],K![-31,-34,38,39,-6,-7],K![21,34,-9,-29,0,5]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-3a)\) = \((a)\cdot(-a^4+4a^2-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 26 \) = \(2\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4a^5-6a^4-19a^3+27a^2+16a-20)\) = \((a)^{10}\cdot(-a^4+4a^2-3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 173056 \) = \(2^{10}\cdot13^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7521611163219}{173056} a^{5} + \frac{7034748606801}{86528} a^{4} + \frac{9426151102851}{86528} a^{3} - \frac{2138246823605}{13312} a^{2} - \frac{4150546552049}{86528} a + \frac{8087484844057}{173056} \)
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 $\left(a^{4} - 5 a^{2} - a + 4 : a^{3} - 5 a - 1 : 1\right)$
Height \(0.023928503438372266921775388857927976285\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{3}{4} a^{5} + \frac{1}{2} a^{4} - \frac{9}{2} a^{3} - \frac{11}{4} a^{2} + \frac{9}{2} a + \frac{3}{4} : \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{5}{8} a^{2} - \frac{9}{4} a - \frac{19}{8} : 1\right)$ $\left(\frac{3}{4} a^{5} + \frac{1}{4} a^{4} - \frac{7}{2} a^{3} - \frac{9}{4} a^{2} + \frac{5}{4} a + \frac{5}{2} : \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{5}{4} a^{3} + \frac{5}{8} a^{2} - \frac{9}{4} a - 3 : 1\right)$
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: \( 0.023928503438372266921775388857927976285 \)
Period: \( 111740.12080240142490056320662020394148 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.93369 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((-a^4+4a^2-3)\) \(13\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 26.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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