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([0,-4,-3,5,1,-1]),K([6,-9,-14,13,4,-3]),K([-2,1,1,0,0,0]),K([3,-3,-4,3,-1,0]),K([3,-10,-12,11,6,-4])])
gp: E = ellinit([Polrev([0,-4,-3,5,1,-1]),Polrev([6,-9,-14,13,4,-3]),Polrev([-2,1,1,0,0,0]),Polrev([3,-3,-4,3,-1,0]),Polrev([3,-10,-12,11,6,-4])], K);
magma: E := EllipticCurve([K![0,-4,-3,5,1,-1],K![6,-9,-14,13,4,-3],K![-2,1,1,0,0,0],K![3,-3,-4,3,-1,0],K![3,-10,-12,11,6,-4]]);
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: | \( -\frac{14540938647674287}{1681} a^{5} + \frac{55342741171488873}{1681} a^{4} - \frac{41785489535847104}{1681} a^{3} - \frac{40862008511943361}{1681} a^{2} + \frac{44715082991653178}{1681} a - \frac{8051386985634292}{1681} \) | ||
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 | $\left(2 a^{5} - 3 a^{4} - 10 a^{3} + 11 a^{2} + 10 a - 3 : 8 a^{5} - 10 a^{4} - 39 a^{3} + 36 a^{2} + 39 a - 11 : 1\right)$ |
| Height | \(0.017230132827428911508020850546813865629\) |
| 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: | $\left(\frac{3}{4} a^{5} - \frac{3}{2} a^{4} - \frac{15}{4} a^{3} + \frac{23}{4} a^{2} + \frac{13}{4} a - \frac{7}{4} : \frac{11}{8} a^{5} - \frac{11}{8} a^{4} - \frac{51}{8} a^{3} + 5 a^{2} + \frac{19}{4} a - \frac{7}{8} : 1\right)$ |
| 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: | \( 0.017230132827428911508020850546813865629 \) | ||
| Period: | \( 32440.583247958131610291615723319297318 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.40753 \) | ||
| 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
41.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.