Minimal equation
Minimal equation
Simplified equation
| $y^2 + y = -28x^6 - 86x^5 - 57x^4 + 45x^3 + 49x^2 - x - 7$ | (homogenize, simplify) |
| $y^2 + z^3y = -28x^6 - 86x^5z - 57x^4z^2 + 45x^3z^3 + 49x^2z^4 - xz^5 - 7z^6$ | (dehomogenize, simplify) |
| $y^2 = -112x^6 - 344x^5 - 228x^4 + 180x^3 + 196x^2 - 4x - 27$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, -1, 49, 45, -57, -86, -28]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-7, -1, 49, 45, -57, -86, -28], R![1]);
sage: X = HyperellipticCurve(R([-27, -4, 196, 180, -228, -344, -112]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(65280\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(130560\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(14918\) | \(=\) | \( 2 \cdot 7459 \) |
| \( I_4 \) | \(=\) | \(13717618\) | \(=\) | \( 2 \cdot 59 \cdot 101 \cdot 1151 \) |
| \( I_6 \) | \(=\) | \(51277691208\) | \(=\) | \( 2^{3} \cdot 3 \cdot 2136570467 \) |
| \( I_{10} \) | \(=\) | \(510\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 17 \) |
| \( J_2 \) | \(=\) | \(29836\) | \(=\) | \( 2^{2} \cdot 7459 \) |
| \( J_4 \) | \(=\) | \(510806\) | \(=\) | \( 2 \cdot 29 \cdot 8807 \) |
| \( J_6 \) | \(=\) | \(8752420\) | \(=\) | \( 2^{2} \cdot 5 \cdot 23 \cdot 53 \cdot 359 \) |
| \( J_8 \) | \(=\) | \(53608371\) | \(=\) | \( 3 \cdot 1093 \cdot 16349 \) |
| \( J_{10} \) | \(=\) | \(130560\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \cdot 17 \) |
| \( g_1 \) | \(=\) | \(46177777988075362598/255\) | ||
| \( g_2 \) | \(=\) | \(105990713496087337/1020\) | ||
| \( g_3 \) | \(=\) | \(24347779975901/408\) |
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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| \(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| \(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 + 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
| \(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
2-torsion field: 8.8.277102632960000.10
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(4\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.917392 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 4.128265 \) |
| Analytic order of Ш: | \( 36 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(8\) | \(9\) | \(2\) | \(1 - 2 T + 2 T^{2}\) | |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) |  |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 5 T^{2} )\) | |
| \(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 17 T^{2} )\) | |
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? |
|---|---|---|
| \(2\) | 2.90.6 | 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);