Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 - 8x^5 - 16x^4 + 25x^3 - 40x^2 + 31x - 8$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - 8x^5z - 16x^4z^2 + 25x^3z^3 - 40x^2z^4 + 31xz^5 - 8z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 32x^5 - 62x^4 + 102x^3 - 159x^2 + 126x - 31$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 31, -40, 25, -16, -8, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-8, 31, -40, 25, -16, -8, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-31, 126, -159, 102, -62, -32, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(5329\) | \(=\) | \( 73^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(5329\) | \(=\) | \( 73^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(139452\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11621 \) |
| \( I_4 \) | \(=\) | \(310895097\) | \(=\) | \( 3 \cdot 103 \cdot 1006133 \) |
| \( I_6 \) | \(=\) | \(16328330150691\) | \(=\) | \( 3 \cdot 13 \cdot 257 \cdot 1629086117 \) |
| \( I_{10} \) | \(=\) | \(-682112\) | \(=\) | \( - 2^{7} \cdot 73^{2} \) |
| \( J_2 \) | \(=\) | \(34863\) | \(=\) | \( 3 \cdot 11621 \) |
| \( J_4 \) | \(=\) | \(37688903\) | \(=\) | \( 7 \cdot 37 \cdot 145517 \) |
| \( J_6 \) | \(=\) | \(-3247242817\) | \(=\) | \( - 7 \cdot 13 \cdot 379 \cdot 94153 \) |
| \( J_8 \) | \(=\) | \(-383415508918120\) | \(=\) | \( - 2^{3} \cdot 5 \cdot 7 \cdot 1369341103279 \) |
| \( J_{10} \) | \(=\) | \(-5329\) | \(=\) | \( - 73^{2} \) |
| \( g_1 \) | \(=\) | \(-51501962646275676450543/5329\) | ||
| \( g_2 \) | \(=\) | \(-1597010473992743939241/5329\) | ||
| \( g_3 \) | \(=\) | \(3946792339710402273/5329\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Mordell-Weil group of the Jacobian
Group structure: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.296753 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 1.187013 \) |
| Analytic order of Ш: | \( 4 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(73\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |  |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.12.2 | no |
| \(3\) | 3.2160.23 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{13}}{2}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{13}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);