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([-3,1,5,0,-1]),K([0,-4,4,1,-1]),K([0,0,0,0,0]),K([90,-333,17,51,-28]),K([-791,2115,3008,-796,-855])])
gp: E = ellinit([Polrev([-3,1,5,0,-1]),Polrev([0,-4,4,1,-1]),Polrev([0,0,0,0,0]),Polrev([90,-333,17,51,-28]),Polrev([-791,2115,3008,-796,-855])], K);
magma: E := EllipticCurve([K![-3,1,5,0,-1],K![0,-4,4,1,-1],K![0,0,0,0,0],K![90,-333,17,51,-28],K![-791,2115,3008,-796,-855]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a-1)\) | = | \((a^3-3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a^4+4a^3-21a^2-12a+1)\) | = | \((a^3-3a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 83521 \) | = | \(17^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{191010208049176061567432964418935750417}{83521} a^{4} + \frac{389373409648192880822797404169052473649}{83521} a^{3} - \frac{161315178223120659347331199752509351188}{83521} a^{2} - \frac{328840231180583481626805905878816700787}{83521} a + \frac{93701569431652774399214957840089814633}{83521} \) | ||
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 | $\left(-\frac{75823468591935}{28059766339609} a^{4} - \frac{79447216842155}{28059766339609} a^{3} + \frac{267777064770760}{28059766339609} a^{2} + \frac{202925795040003}{28059766339609} a + \frac{56944293685483}{28059766339609} : \frac{796278179042250939088}{148636707086560795523} a^{4} + \frac{2624872917322746219746}{148636707086560795523} a^{3} - \frac{3262843028579578819662}{148636707086560795523} a^{2} - \frac{10855781245656676923987}{148636707086560795523} a + \frac{2395335271138431630239}{148636707086560795523} : 1\right)$ |
| Height | \(5.2256210355640711675946979979914220240\) |
| 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: | $\left(\frac{9}{2} a^{4} + \frac{11}{4} a^{3} - \frac{87}{4} a^{2} - \frac{61}{4} a + \frac{25}{4} : -\frac{1}{2} a^{4} + \frac{1}{2} a^{3} + \frac{17}{4} a^{2} + \frac{21}{8} a - \frac{1}{2} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 5.2256210355640711675946979979914220240 \) | ||
| Period: | \( 6.0005706044630956041090036444709045513 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.59665569 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a-1)\) | \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
17.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.