Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = x^6 - 9x^5 + x^4 + 40x^3 - 2x^2 - 51x - 23$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = x^6 - 9x^5z + x^4z^2 + 40x^3z^3 - 2x^2z^4 - 51xz^5 - 23z^6$ | (dehomogenize, simplify) |
| $y^2 = 5x^6 - 36x^5 + 4x^4 + 162x^3 - 8x^2 - 204x - 91$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-23, -51, -2, 40, 1, -9, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-23, -51, -2, 40, 1, -9, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-91, -204, -8, 162, 4, -36, 5]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(121077\) | \(=\) | \( 3^{2} \cdot 11 \cdot 1223 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(121077\) | \(=\) | \( 3^{2} \cdot 11 \cdot 1223 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(280468\) | \(=\) | \( 2^{2} \cdot 70117 \) |
| \( I_4 \) | \(=\) | \(202372345\) | \(=\) | \( 5 \cdot 7 \cdot 1399 \cdot 4133 \) |
| \( I_6 \) | \(=\) | \(17960645748237\) | \(=\) | \( 3 \cdot 5986881916079 \) |
| \( I_{10} \) | \(=\) | \(15497856\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 11 \cdot 1223 \) |
| \( J_2 \) | \(=\) | \(70117\) | \(=\) | \( 70117 \) |
| \( J_4 \) | \(=\) | \(196417556\) | \(=\) | \( 2^{2} \cdot 2833 \cdot 17333 \) |
| \( J_6 \) | \(=\) | \(712749098088\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 22063 \cdot 149561 \) |
| \( J_8 \) | \(=\) | \(2848993051405790\) | \(=\) | \( 2 \cdot 5 \cdot 452821 \cdot 629165399 \) |
| \( J_{10} \) | \(=\) | \(121077\) | \(=\) | \( 3^{2} \cdot 11 \cdot 1223 \) |
| \( g_1 \) | \(=\) | \(1694792881814644976830357/121077\) | ||
| \( g_2 \) | \(=\) | \(67709605216733368757828/121077\) | ||
| \( g_3 \) | \(=\) | \(389350574186698351848/13453\) |
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 except over $\Q_{3}$.
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: \(\Z/{2}\Z\) (conditional)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 6xz - 7z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(-71xz^2 - 67z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 6xz - 7z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(-71xz^2 - 67z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(5x^2 - 6xz - 7z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(x^3 - 142xz^2 - 133z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.4586838448896.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) (lower bound) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 1.0 \) |
| Real period: | \( 1.324041 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 5.296165 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |  |
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 11 T^{2} )\) |  |
| \(1223\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 32 T + 1223 T^{2} )\) | |
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? |
|---|---|---|
| \(2\) | 2.15.1 | 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);