Base field 5.5.149169.1
Generator \(a\), with minimal polynomial \( x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([1, 4, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,-1,-10,-1,2]),K([-5,4,2,-1,0]),K([2,-3,-1,1,0]),K([-41,-204,-147,23,27]),K([-236,-1241,-950,132,175])])
gp: E = ellinit([Polrev([4,-1,-10,-1,2]),Polrev([-5,4,2,-1,0]),Polrev([2,-3,-1,1,0]),Polrev([-41,-204,-147,23,27]),Polrev([-236,-1241,-950,132,175])], K);
magma: E := EllipticCurve([K![4,-1,-10,-1,2],K![-5,4,2,-1,0],K![2,-3,-1,1,0],K![-41,-204,-147,23,27],K![-236,-1241,-950,132,175]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+a^2+4a-1)\) | = | \((-a^4+a^3+4a^2-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^4+3a^3+20a^2-7)\) | = | \((-a^4+a^3+4a^2-1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -2187 \) | = | \(-3^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{48552525146}{3} a^{4} + \frac{36513739424}{3} a^{3} - \frac{263855133073}{3} a^{2} - \frac{344088812371}{3} a - \frac{64560563147}{3} \) | ||
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: | \(0\) |
| 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 241.29423160411227936595884836674893858 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.24950399 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+4a^2-1)\) | \(3\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
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 9.1-b 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.