Base field 6.6.1279733.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0,0,0]),K([0,3,1,-1,0,0]),K([1,0,0,0,0,0]),K([-11,18,6,-6,0,0]),K([16,2,-18,-2,4,0])])
gp: E = ellinit([Polrev([1,0,0,0,0,0]),Polrev([0,3,1,-1,0,0]),Polrev([1,0,0,0,0,0]),Polrev([-11,18,6,-6,0,0]),Polrev([16,2,-18,-2,4,0])], K);
magma: E := EllipticCurve([K![1,0,0,0,0,0],K![0,3,1,-1,0,0],K![1,0,0,0,0,0],K![-11,18,6,-6,0,0],K![16,2,-18,-2,4,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-2a^4-3a^3+6a^2-4)\) | = | \((a^5-2a^4-3a^3+6a^2-4)\) |
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: | \((3a^4+4a^3-19a^2-15a+9)\) | = | \((a^5-2a^4-3a^3+6a^2-4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2825761 \) | = | \(41^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2962060985575}{1681} a^{4} + \frac{731644682050}{1681} a^{3} - \frac{15541949609925}{1681} a^{2} - \frac{5156995031725}{1681} a + \frac{14955417009784}{1681} \) | ||
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: | \(1\) |
| Generator | $\left(-\frac{25}{13} a^{4} + \frac{46}{13} a^{3} + \frac{79}{13} a^{2} - \frac{113}{13} a + \frac{16}{13} : -\frac{478}{169} a^{5} + \frac{950}{169} a^{4} + \frac{1202}{169} a^{3} - \frac{2465}{169} a^{2} + \frac{1242}{169} a - \frac{125}{169} : 1\right)$ |
| Height | \(0.45612644674443973181760784589572129183\) |
| Torsion structure: | \(\Z/10\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(0 : -2 a^{3} + 2 a^{2} + 6 a - 4 : 1\right)$ |
| 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: | \( 0.45612644674443973181760784589572129183 \) | ||
| Period: | \( 28807.736818103481384910484210573744088 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(10\) | ||
| Leading coefficient: | \( 2.78770 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-2a^4-3a^3+6a^2-4)\) | \(41\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
41.1-b
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\zeta_{7})^+\) | a curve with conductor norm 284089 (not in the database) |
| \(\Q(\zeta_{7})^+\) | 3.3.49.1-41.2-a2 |