Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,9,10,-11,-2,2]),K([-1,-3,-4,5,1,-1]),K([-2,17,10,-17,-2,3]),K([-16,92,47,-106,-10,19]),K([-40,286,137,-312,-29,55])])
gp: E = ellinit([Polrev([-4,9,10,-11,-2,2]),Polrev([-1,-3,-4,5,1,-1]),Polrev([-2,17,10,-17,-2,3]),Polrev([-16,92,47,-106,-10,19]),Polrev([-40,286,137,-312,-29,55])], K);
magma: E := EllipticCurve([K![-4,9,10,-11,-2,2],K![-1,-3,-4,5,1,-1],K![-2,17,10,-17,-2,3],K![-16,92,47,-106,-10,19],K![-40,286,137,-312,-29,55]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-6a^3+4a^2+7a+1)\) | = | \((a^5-a^4-6a^3+4a^2+7a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+5a^3-5a+2)\) | = | \((a^5-a^4-6a^3+4a^2+7a+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1681 \) | = | \(-41^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{46785115689974520}{1681} a^{5} + \frac{25523470799745310}{1681} a^{4} + \frac{267495305548152369}{1681} a^{3} - \frac{124251905123044313}{1681} a^{2} - \frac{253745891578685595}{1681} a + \frac{46114837230979857}{1681} \) | ||
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: | $\left(\frac{9}{2} a^{5} - \frac{11}{4} a^{4} - \frac{105}{4} a^{3} + \frac{53}{4} a^{2} + \frac{107}{4} a - \frac{19}{4} : -\frac{23}{8} a^{5} + \frac{3}{2} a^{4} + \frac{137}{8} a^{3} - 7 a^{2} - \frac{155}{8} a + \frac{1}{8} : 1\right)$ |
| 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: | \( 584.78187354006384422931131082328955483 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.39442 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-a^4-6a^3+4a^2+7a+1)\) | \(41\) | \(2\) | \(I_{2}\) | Non-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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
41.4-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.