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([1,-3,-4,1,1]),K([1,0,0,0,0]),K([1,-3,-4,1,1]),K([265,-755,-995,180,220]),K([3361,-9228,-11769,2159,2642])])
gp: E = ellinit([Polrev([1,-3,-4,1,1]),Polrev([1,0,0,0,0]),Polrev([1,-3,-4,1,1]),Polrev([265,-755,-995,180,220]),Polrev([3361,-9228,-11769,2159,2642])], K);
magma: E := EllipticCurve([K![1,-3,-4,1,1],K![1,0,0,0,0],K![1,-3,-4,1,1],K![265,-755,-995,180,220],K![3361,-9228,-11769,2159,2642]]);
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: | \((-489a^4-856a^3-1899a^2-380a+11167)\) | = | \((a^3-3a-1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -48661191875666868481 \) | = | \(-17^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4610174704839696307681031019985}{48661191875666868481} a^{4} - \frac{3646564185715492789590081314801}{48661191875666868481} a^{3} + \frac{20165287106639313522557968182292}{48661191875666868481} a^{2} + \frac{15946358590737268511990785275315}{48661191875666868481} a - \frac{5829599632074986762576303718745}{48661191875666868481} \) | ||
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: | $\left(\frac{9}{2} a^{4} + \frac{17}{4} a^{3} - \frac{75}{4} a^{2} - 14 a + \frac{7}{4} : \frac{25}{4} a^{4} + \frac{35}{8} a^{3} - \frac{61}{2} a^{2} - \frac{183}{8} a + \frac{67}{8} : 1\right)$ |
| 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: | \( 3.7310507290234964362871298029418305363 \) | ||
| Tamagawa product: | \( 16 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.21588343 \) | ||
| Analytic order of Ш: | \( 16 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a-1)\) | \(17\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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-b
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.