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([-1,5,4,-5,-1,1]),K([-4,0,9,-4,-2,1]),K([-3,4,5,-5,-1,1]),K([12,-8,-167,85,38,-19]),K([129,53,-942,473,197,-105])])
gp: E = ellinit([Polrev([-1,5,4,-5,-1,1]),Polrev([-4,0,9,-4,-2,1]),Polrev([-3,4,5,-5,-1,1]),Polrev([12,-8,-167,85,38,-19]),Polrev([129,53,-942,473,197,-105])], K);
magma: E := EllipticCurve([K![-1,5,4,-5,-1,1],K![-4,0,9,-4,-2,1],K![-3,4,5,-5,-1,1],K![12,-8,-167,85,38,-19],K![129,53,-942,473,197,-105]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-4a^3+4a^2-2)\) | = | \((a^5-a^4-4a^3+4a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 47 \) | = | \(47\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+a^4+4a^3-4a^2+2)\) | = | \((a^5-a^4-4a^3+4a^2-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -47 \) | = | \(-47\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{21417905398977605}{47} a^{5} + \frac{62004666093510408}{47} a^{4} - \frac{10624618770251898}{47} a^{3} - \frac{65152171397191321}{47} a^{2} + \frac{16575370430512357}{47} a + \frac{11272985421638474}{47} \) | ||
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 | $\left(19 a^{5} - 9 a^{4} - 96 a^{3} + 24 a^{2} + 86 a + 22 : 110 a^{5} - 46 a^{4} - 563 a^{3} + 91 a^{2} + 562 a + 157 : 1\right)$ |
| Height | \(1.2799600035188543476811824635816111570\) |
| 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: | $\left(4 a^{5} - 4 a^{4} - 16 a^{3} + 11 a^{2} + 6 a + 3 : -4 a^{5} + 2 a^{4} + 20 a^{3} - 6 a^{2} - 18 a : 1\right)$ |
| 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: | \( 1.2799600035188543476811824635816111570 \) | ||
| Period: | \( 11.565787949700540708792798169997877455 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.33639 \) | ||
| Analytic order of Ш: | \( 81 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-a^4-4a^3+4a^2-2)\) | \(47\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
47.1-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.