Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,4,0,-5,0,1]),K([2,-5,-9,8,3,-2]),K([-1,-2,0,1,0,0]),K([18,-118,-89,107,28,-22]),K([138,-420,-397,399,112,-82])])
gp: E = ellinit([Polrev([1,4,0,-5,0,1]),Polrev([2,-5,-9,8,3,-2]),Polrev([-1,-2,0,1,0,0]),Polrev([18,-118,-89,107,28,-22]),Polrev([138,-420,-397,399,112,-82])], K);
magma: E := EllipticCurve([K![1,4,0,-5,0,1],K![2,-5,-9,8,3,-2],K![-1,-2,0,1,0,0],K![18,-118,-89,107,28,-22],K![138,-420,-397,399,112,-82]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-5a^3+5a)\) | = | \((a^5-5a^3+5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^5-2a^4-12a^3+9a^2+16a-8)\) | = | \((a^5-5a^3+5a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1681 \) | = | \(-41^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{247312191821393399}{1681} a^{5} + \frac{434315726526349312}{1681} a^{4} + \frac{1095159425055437131}{1681} a^{3} - \frac{1711435916724291643}{1681} a^{2} - \frac{911968936218596275}{1681} a + \frac{1014171519640582545}{1681} \) | ||
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 | $\left(-3 a^{5} + 3 a^{4} + 15 a^{3} - 11 a^{2} - 15 a + 6 : 4 a^{5} - 4 a^{4} - 20 a^{3} + 14 a^{2} + 18 a - 6 : 1\right)$ |
| Height | \(0.026734693097681236172118651495903639808\) |
| 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: | $\left(-\frac{9}{4} a^{5} + \frac{11}{4} a^{4} + \frac{23}{2} a^{3} - 11 a^{2} - \frac{51}{4} a + \frac{11}{2} : 3 a^{5} - \frac{13}{4} a^{4} - \frac{61}{4} a^{3} + 13 a^{2} + \frac{123}{8} a - \frac{43}{8} : 1\right)$ |
| 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: | \( 0.026734693097681236172118651495903639808 \) | ||
| Period: | \( 19378.268634232349756069043160243624743 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.23144 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-5a^3+5a)\) | \(41\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
41.1-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.