Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,4,5,-5,-1,1]),K([2,-1,-8,4,2,-1]),K([-4,6,5,-9,-1,2]),K([7,6,-23,-10,5,2]),K([6,-17,-42,84,13,-21])])
gp: E = ellinit([Polrev([-4,4,5,-5,-1,1]),Polrev([2,-1,-8,4,2,-1]),Polrev([-4,6,5,-9,-1,2]),Polrev([7,6,-23,-10,5,2]),Polrev([6,-17,-42,84,13,-21])], K);
magma: E := EllipticCurve([K![-4,4,5,-5,-1,1],K![2,-1,-8,4,2,-1],K![-4,6,5,-9,-1,2],K![7,6,-23,-10,5,2],K![6,-17,-42,84,13,-21]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+a^4+9a^3-4a^2-6a+1)\) | = | \((-2a^5+a^4+9a^3-4a^2-6a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 83 \) | = | \(83\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4a^5-5a^4-29a^3+24a^2+43a-18)\) | = | \((-2a^5+a^4+9a^3-4a^2-6a+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 47458321 \) | = | \(83^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{15736963093640}{47458321} a^{5} - \frac{44917566551496}{47458321} a^{4} + \frac{6514884455608}{47458321} a^{3} + \frac{47332089860599}{47458321} a^{2} - \frac{11320884612420}{47458321} a - \frac{8429605989992}{47458321} \) | ||
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 | $\left(5 a^{5} - 10 a^{4} - 17 a^{3} + 38 a^{2} - 4 a - 8 : 27 a^{5} - 52 a^{4} - 92 a^{3} + 194 a^{2} - 25 a - 34 : 1\right)$ |
| Height | \(0.0077073270358283725055118910388713102221\) |
| 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: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0077073270358283725055118910388713102221 \) | ||
| Period: | \( 12401.088435760592188712674491098456689 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.97969 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+9a^3-4a^2-6a+1)\) | \(83\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 83.1-c 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.