Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,5,4,-9,-1,2]),K([3,0,-1,0,0,0]),K([-4,5,5,-9,-1,2]),K([-5,-22,9,26,-2,-5]),K([-9,-10,56,-4,-14,2])])
gp: E = ellinit([Polrev([-3,5,4,-9,-1,2]),Polrev([3,0,-1,0,0,0]),Polrev([-4,5,5,-9,-1,2]),Polrev([-5,-22,9,26,-2,-5]),Polrev([-9,-10,56,-4,-14,2])], K);
magma: E := EllipticCurve([K![-3,5,4,-9,-1,2],K![3,0,-1,0,0,0],K![-4,5,5,-9,-1,2],K![-5,-22,9,26,-2,-5],K![-9,-10,56,-4,-14,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 73 \) | = | \(73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^5+20a^4-9a^3-79a^2+49a-46)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 151334226289 \) | = | \(73^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1750397764286997144}{151334226289} a^{5} + \frac{2384110649186678931}{151334226289} a^{4} + \frac{7888305624092684601}{151334226289} a^{3} - \frac{9728289241621098752}{151334226289} a^{2} - \frac{5474432982410677682}{151334226289} a + \frac{5421863886826175453}{151334226289} \) | ||
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 | $\left(3 a^{5} + a^{4} - 14 a^{3} - 3 a^{2} + 13 a + 3 : a^{5} - 5 a^{4} - 17 a^{3} + 4 a^{2} + 27 a + 8 : 1\right)$ |
| Height | \(0.090860474324345564396764870179007434994\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{5} - 2 a^{4} - 3 a^{3} + 8 a^{2} - 2 a - 2 : -2 a^{5} + 6 a^{4} + 4 a^{3} - 24 a^{2} + 10 a + 6 : 1\right)$ |
| 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: | \( 0.090860474324345564396764870179007434994 \) | ||
| Period: | \( 22274.091166549739976824916381143308353 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.62889 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+2a^4+5a^3-8a^2-4a+3)\) | \(73\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
73.2-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.