Minimal equation
Minimal equation
Simplified equation
| $y^2 + xy = 8x^5 - 21x^4 + 15x^3 - x^2 - x$ | (homogenize, simplify) |
| $y^2 + xz^2y = 8x^5z - 21x^4z^2 + 15x^3z^3 - x^2z^4 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = 32x^5 - 84x^4 + 60x^3 - 3x^2 - 4x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 15, -21, 8]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 15, -21, 8], R![0, 1]);
sage: X = HyperellipticCurve(R([0, -4, -3, 60, -84, 32]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(1312\) | \(=\) | \( 2^{5} \cdot 41 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(83968\) | \(=\) | \( 2^{11} \cdot 41 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(1556\) | \(=\) | \( 2^{2} \cdot 389 \) |
| \( I_4 \) | \(=\) | \(36553\) | \(=\) | \( 11 \cdot 3323 \) |
| \( I_6 \) | \(=\) | \(20209667\) | \(=\) | \( 20209667 \) |
| \( I_{10} \) | \(=\) | \(10496\) | \(=\) | \( 2^{8} \cdot 41 \) |
| \( J_2 \) | \(=\) | \(1556\) | \(=\) | \( 2^{2} \cdot 389 \) |
| \( J_4 \) | \(=\) | \(76512\) | \(=\) | \( 2^{5} \cdot 3 \cdot 797 \) |
| \( J_6 \) | \(=\) | \(1289104\) | \(=\) | \( 2^{4} \cdot 23 \cdot 31 \cdot 113 \) |
| \( J_8 \) | \(=\) | \(-962060080\) | \(=\) | \( - 2^{4} \cdot 5 \cdot 41 \cdot 293311 \) |
| \( J_{10} \) | \(=\) | \(83968\) | \(=\) | \( 2^{11} \cdot 41 \) |
| \( g_1 \) | \(=\) | \(8907339520949/82\) | ||
| \( g_2 \) | \(=\) | \(140743510779/41\) | ||
| \( g_3 \) | \(=\) | \(12191781649/328\) |
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
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1)\)
magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 4xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
| \((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(6\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 4xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
| \((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(6\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 4xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
| \((1 : 1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(6\) |
2-torsion field: \(\Q(\sqrt{2}, \sqrt{41})\)
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 14.58470 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.405130 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(5\) | \(11\) | \(4\) | \(1 + T\) | |
| \(41\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 6 T + 41 T^{2} )\) |  |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.180.3 | yes |
| \(3\) | 3.80.1 | yes |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
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);