Base field 5.5.160801.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 4 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,-4,0,1]),K([3,3,-6,-1,1]),K([1,0,0,0,0]),K([5,-2,-14,0,3]),K([7,-13,-37,2,8])])
gp: E = ellinit([Polrev([0,1,-4,0,1]),Polrev([3,3,-6,-1,1]),Polrev([1,0,0,0,0]),Polrev([5,-2,-14,0,3]),Polrev([7,-13,-37,2,8])], K);
magma: E := EllipticCurve([K![0,1,-4,0,1],K![3,3,-6,-1,1],K![1,0,0,0,0],K![5,-2,-14,0,3],K![7,-13,-37,2,8]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-5a^2+4a+1)\) | = | \((-a^4+5a^2+a-3)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^4-3a^3-14a^2+12a-4)\) | = | \((-a^4+5a^2+a-3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 19683 \) | = | \(3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 199781 a^{4} - 165912 a^{3} - 1009767 a^{2} + 603539 a + 629286 \) | ||
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 | $\left(-a^{4} + 5 a^{2} + a - 1 : -3 a^{4} - a^{3} + 13 a^{2} + 4 a - 3 : 1\right)$ |
| Height | \(0.069318596270554109564829055334969216883\) |
| 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: | \( 0.069318596270554109564829055334969216883 \) | ||
| Period: | \( 4244.2502979604471765262814533689300072 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.66839742 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+5a^2+a-3)\) | \(3\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(3\) | \(9\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.3-d
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.