Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-6,-6,5,2,-1]),K([-1,3,2,-1,0,0]),K([0,4,-2,-2,1,0]),K([7,-9,-20,9,8,-2]),K([1,6,-10,-12,4,4])])
gp: E = ellinit([Polrev([3,-6,-6,5,2,-1]),Polrev([-1,3,2,-1,0,0]),Polrev([0,4,-2,-2,1,0]),Polrev([7,-9,-20,9,8,-2]),Polrev([1,6,-10,-12,4,4])], K);
magma: E := EllipticCurve([K![3,-6,-6,5,2,-1],K![-1,3,2,-1,0,0],K![0,4,-2,-2,1,0],K![7,-9,-20,9,8,-2],K![1,6,-10,-12,4,4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-3a^4-3a^3+8a^2+2a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^5-8a^4-10a^3+26a^2+5a-14)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 19683 \) | = | \(3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 9252524 a^{5} - 42006713 a^{4} + 46188002 a^{3} + 12132411 a^{2} - 27928899 a + 6008055 \) | ||
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 | $\left(-11 a^{5} + 30 a^{4} + 30 a^{3} - 91 a^{2} - 14 a + 39 : 61 a^{5} - 167 a^{4} - 164 a^{3} + 506 a^{2} + 69 a - 226 : 1\right)$ |
| Height | \(0.0036517829885721044458552651564831764474\) |
| 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: | \( 0.0036517829885721044458552651564831764474 \) | ||
| Period: | \( 54456.075689713167567984588108295958778 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.03935 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(3\) | \(9\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Ns |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 27.1-n 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.