Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2)y = -x^6 + 22x^5 - 216x^4 + 636x^3 + 72x^2 + 2x$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z)y = -x^6 + 22x^5z - 216x^4z^2 + 636x^3z^3 + 72x^2z^4 + 2xz^5$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 90x^5 - 863x^4 + 2544x^3 + 288x^2 + 8x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 72, 636, -216, 22, -1]), R([0, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 72, 636, -216, 22, -1], R![0, 0, 1, 1]);
sage: X = HyperellipticCurve(R([0, 8, 288, 2544, -863, 90, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(149688\) | \(=\) | \( 2^{3} \cdot 3^{5} \cdot 7 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-299376\) | \(=\) | \( - 2^{4} \cdot 3^{5} \cdot 7 \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(10709280\) | \(=\) | \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 37 \cdot 67 \) |
| \( I_4 \) | \(=\) | \(255376080\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 354689 \) |
| \( I_6 \) | \(=\) | \(910184353854876\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 3223499 \cdot 7843309 \) |
| \( I_{10} \) | \(=\) | \(-1197504\) | \(=\) | \( - 2^{6} \cdot 3^{5} \cdot 7 \cdot 11 \) |
| \( J_2 \) | \(=\) | \(5354640\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 5 \cdot 37 \cdot 67 \) |
| \( J_4 \) | \(=\) | \(1194631167720\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 34963 \cdot 284737 \) |
| \( J_6 \) | \(=\) | \(355353717454751236\) | \(=\) | \( 2^{2} \cdot 13 \cdot 1637 \cdot 4174542049889 \) |
| \( J_8 \) | \(=\) | \(118911900685964597084160\) | \(=\) | \( 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 233 \cdot 1103 \cdot 89652704329 \) |
| \( J_{10} \) | \(=\) | \(-299376\) | \(=\) | \( - 2^{4} \cdot 3^{5} \cdot 7 \cdot 11 \) |
| \( g_1 \) | \(=\) | \(-1132205169121716742666444800000/77\) | ||
| \( g_2 \) | \(=\) | \(-47173534667132877160047360000/77\) | ||
| \( g_3 \) | \(=\) | \(-2620566365698612668618091200/77\) |
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: \((0 : 0 : 1)\)
magma: [C![0,0,1]]; // minimal model
magma: [C![0,0,1]]; // simplified model
Number of rational Weierstrass points: \(1\)
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 18xz - z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-127xz^2 - 7z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 18xz - z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-127xz^2 - 7z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 18xz - z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^3 + x^2z - 254xz^2 - 14z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.816903549.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.439731 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 3.517849 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(3\) | \(4\) | \(2\) | \(1 - T\) | |
| \(3\) | \(5\) | \(5\) | \(1\) | \(1 - T\) |  |
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 7 T^{2} )\) |  |
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 11 T^{2} )\) | |
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? |
|---|---|---|
| \(2\) | 2.60.1 | 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);