Base field 4.4.16997.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - x + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([5, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-1,1]),K([-3,1,1,0]),K([0,1,0,0]),K([-4929,5259,1237,-1147]),K([-163829,176448,42325,-37920])])
gp: E = ellinit([Polrev([2,-3,-1,1]),Polrev([-3,1,1,0]),Polrev([0,1,0,0]),Polrev([-4929,5259,1237,-1147]),Polrev([-163829,176448,42325,-37920])], K);
magma: E := EllipticCurve([K![2,-3,-1,1],K![-3,1,1,0],K![0,1,0,0],K![-4929,5259,1237,-1147],K![-163829,176448,42325,-37920]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2-a+1)\) | = | \((-a^2-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-5a-3)\) | = | \((-a^2-a+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{5357496973442499987}{25} a^{3} + \frac{4190018497495156254}{25} a^{2} - \frac{29759807723130703029}{25} a - \frac{6303903372448402516}{5} \) | ||
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 | $\left(\frac{1931311047267}{166430300} a^{3} - \frac{2208580773749}{166430300} a^{2} - \frac{9053260565551}{166430300} a + \frac{4218048007391}{83215150} : -\frac{1342271504794492969}{411881706440} a^{3} + \frac{19206572057197366679}{5148521330500} a^{2} + \frac{6295050386739555759}{411881706440} a - \frac{73293323037603288731}{5148521330500} : 1\right)$ |
| Height | \(13.543810162753335093940415076862624058\) |
| 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: | \( 13.543810162753335093940415076862624058 \) | ||
| Period: | \( 0.13895377824000150379082923899943906993 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.88705512498013 \) | ||
| Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2-a+1)\) | \(5\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
5.2-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.