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
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.