Base field 6.6.1397493.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,4,-3,-2,1,0]),K([-1,9,0,-4,1,0]),K([-2,9,6,-5,-2,1]),K([22,-89,-73,51,22,-10]),K([54,-220,-286,142,99,-38])])
gp: E = ellinit([Polrev([2,4,-3,-2,1,0]),Polrev([-1,9,0,-4,1,0]),Polrev([-2,9,6,-5,-2,1]),Polrev([22,-89,-73,51,22,-10]),Polrev([54,-220,-286,142,99,-38])], K);
magma: E := EllipticCurve([K![2,4,-3,-2,1,0],K![-1,9,0,-4,1,0],K![-2,9,6,-5,-2,1],K![22,-89,-73,51,22,-10],K![54,-220,-286,142,99,-38]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+2a^3+3a^2-3a-2)\) | = | \((-a^4+2a^3+3a^2-3a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 37 \) | = | \(37\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^5+5a^4+19a^3-19a^2-29a+11)\) | = | \((-a^4+2a^3+3a^2-3a-2)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 50653 \) | = | \(37^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{13002440326011}{50653} a^{5} + \frac{36425460538638}{50653} a^{4} + \frac{46237058672607}{50653} a^{3} - \frac{120843085936113}{50653} a^{2} - \frac{62990925595713}{50653} a + \frac{65510248079025}{50653} \) | ||
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 | $\left(-a^{5} + 2 a^{4} + 5 a^{3} - 5 a^{2} - 10 a + 1 : a^{5} - 6 a^{4} + 9 a^{3} + 6 a^{2} - 19 a + 5 : 1\right)$ |
| Height | \(0.21667936921298793038676538439878777631\) |
| 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: | \( 0.21667936921298793038676538439878777631 \) | ||
| Period: | \( 692.64729843986984605309379100841358388 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.28521 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+2a^3+3a^2-3a-2)\) | \(37\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
37.1-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.