Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x + 1)y = -11x^6 + 6x^5 + 22x^4 - 16x^3 - 6x^2 + 11x - 5$ | (homogenize, simplify) |
| $y^2 + (xz^2 + z^3)y = -11x^6 + 6x^5z + 22x^4z^2 - 16x^3z^3 - 6x^2z^4 + 11xz^5 - 5z^6$ | (dehomogenize, simplify) |
| $y^2 = -44x^6 + 24x^5 + 88x^4 - 64x^3 - 23x^2 + 46x - 19$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 11, -6, -16, 22, 6, -11]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 11, -6, -16, 22, 6, -11], R![1, 1]);
sage: X = HyperellipticCurve(R([-19, 46, -23, -64, 88, 24, -44]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(4336\) | \(=\) | \( 2^{4} \cdot 271 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(138752\) | \(=\) | \( 2^{9} \cdot 271 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(12440\) | \(=\) | \( 2^{3} \cdot 5 \cdot 311 \) |
| \( I_4 \) | \(=\) | \(748636\) | \(=\) | \( 2^{2} \cdot 7 \cdot 26737 \) |
| \( I_6 \) | \(=\) | \(21969201218\) | \(=\) | \( 2 \cdot 17 \cdot 20533 \cdot 31469 \) |
| \( I_{10} \) | \(=\) | \(-17344\) | \(=\) | \( - 2^{6} \cdot 271 \) |
| \( J_2 \) | \(=\) | \(12440\) | \(=\) | \( 2^{3} \cdot 5 \cdot 311 \) |
| \( J_4 \) | \(=\) | \(5948976\) | \(=\) | \( 2^{4} \cdot 3 \cdot 11 \cdot 19 \cdot 593 \) |
| \( J_6 \) | \(=\) | \(-13347212816\) | \(=\) | \( - 2^{4} \cdot 7 \cdot 41 \cdot 2906623 \) |
| \( J_8 \) | \(=\) | \(-50357410719904\) | \(=\) | \( - 2^{5} \cdot 181 \cdot 193 \cdot 45048209 \) |
| \( J_{10} \) | \(=\) | \(-138752\) | \(=\) | \( - 2^{9} \cdot 271 \) |
| \( g_1 \) | \(=\) | \(-581878004510200000/271\) | ||
| \( g_2 \) | \(=\) | \(-22368321536682000/271\) | ||
| \( g_3 \) | \(=\) | \(4034236783676050/271\) |
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 $\Q_{2}$.
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: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.376283 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.752566 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(9\) | \(1\) | \(1 + T\) | |
| \(271\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 271 T^{2} )\) |  |
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? |
|---|---|---|
| \(3\) | 3.80.2 | no |
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);