Base field 5.5.38569.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,-3,-9,1,2]),K([3,0,-9,0,2]),K([-2,1,1,0,0]),K([18,-14,-27,2,6]),K([6,-10,12,14,-9])])
gp: E = ellinit([Polrev([5,-3,-9,1,2]),Polrev([3,0,-9,0,2]),Polrev([-2,1,1,0,0]),Polrev([18,-14,-27,2,6]),Polrev([6,-10,12,14,-9])], K);
magma: E := EllipticCurve([K![5,-3,-9,1,2],K![3,0,-9,0,2],K![-2,1,1,0,0],K![18,-14,-27,2,6],K![6,-10,12,14,-9]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+6a^2+2a-3)\) | = | \((-a^4+6a^2+2a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^4+a^3+4a^2-2a-2)\) | = | \((-a^4+6a^2+2a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 43 \) | = | \(43\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{51665932627}{43} a^{4} - \frac{10018445802}{43} a^{3} + \frac{250693585412}{43} a^{2} + \frac{53592431783}{43} a - \frac{173974818865}{43} \) | ||
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 | $\left(-a^{4} - a^{3} + 5 a^{2} + 2 a - 2 : a^{3} - a^{2} - 2 a + 3 : 1\right)$ |
| Height | \(0.0063189158403372600625030208892289097783\) |
| 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: | \( 0.0063189158403372600625030208892289097783 \) | ||
| Period: | \( 12588.521348623520338722331042412597325 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.02520088 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+6a^2+2a-3)\) | \(43\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 43.2-d 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.