Base field 5.5.70601.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,0,-10,-1,2]),K([-1,3,1,-1,0]),K([4,0,-10,-1,2]),K([-33769,-3366,56621,7306,-10860]),K([-2231590,-228180,3752939,486907,-719344])])
gp: E = ellinit([Polrev([4,0,-10,-1,2]),Polrev([-1,3,1,-1,0]),Polrev([4,0,-10,-1,2]),Polrev([-33769,-3366,56621,7306,-10860]),Polrev([-2231590,-228180,3752939,486907,-719344])], K);
magma: E := EllipticCurve([K![4,0,-10,-1,2],K![-1,3,1,-1,0],K![4,0,-10,-1,2],K![-33769,-3366,56621,7306,-10860],K![-2231590,-228180,3752939,486907,-719344]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+a^2+4a)\) | = | \((-a^3+a^2+4a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^4-a^3-10a^2+4)\) | = | \((-a^3+a^2+4a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 81 \) | = | \(9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1241019728491478667854287148915537}{9} a^{4} - \frac{3466470975193904516888121233549840}{9} a^{3} + \frac{11129815715759114365727588897188}{9} a^{2} + \frac{2462080981149287745534356743321450}{9} a - \frac{692052889851961102412512823946311}{9} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{143}{2} a^{4} - \frac{203}{4} a^{3} - \frac{1499}{4} a^{2} + \frac{65}{2} a + \frac{899}{4} : -\frac{889}{8} a^{4} + \frac{603}{8} a^{3} + \frac{4661}{8} a^{2} - \frac{215}{8} a - \frac{1371}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.27198384054041457956793812673780906033 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.27952180 \) | ||
| Analytic order of Ш: | \( 2500 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.