Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x + 1)y = -x^6 - 4x^5 + 4x^4 + 23x^3 - 21x^2 - 9x - 1$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2 + z^3)y = -x^6 - 4x^5z + 4x^4z^2 + 23x^3z^3 - 21x^2z^4 - 9xz^5 - z^6$ | (dehomogenize, simplify) |
| $y^2 = -4x^6 - 16x^5 + 17x^4 + 94x^3 - 81x^2 - 34x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -9, -21, 23, 4, -4, -1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -9, -21, 23, 4, -4, -1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-3, -34, -81, 94, 17, -16, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(32119\) | \(=\) | \( 32119 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(32119\) | \(=\) | \( 32119 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(46964\) | \(=\) | \( 2^{2} \cdot 59 \cdot 199 \) |
| \( I_4 \) | \(=\) | \(147959497\) | \(=\) | \( 7 \cdot 31 \cdot 681841 \) |
| \( I_6 \) | \(=\) | \(1716344218693\) | \(=\) | \( 13 \cdot 17 \cdot 19 \cdot 408750707 \) |
| \( I_{10} \) | \(=\) | \(4111232\) | \(=\) | \( 2^{7} \cdot 32119 \) |
| \( J_2 \) | \(=\) | \(11741\) | \(=\) | \( 59 \cdot 199 \) |
| \( J_4 \) | \(=\) | \(-421184\) | \(=\) | \( - 2^{6} \cdot 6581 \) |
| \( J_6 \) | \(=\) | \(14829864\) | \(=\) | \( 2^{3} \cdot 3 \cdot 7 \cdot 41 \cdot 2153 \) |
| \( J_8 \) | \(=\) | \(-819632158\) | \(=\) | \( - 2 \cdot 409816079 \) |
| \( J_{10} \) | \(=\) | \(32119\) | \(=\) | \( 32119 \) |
| \( g_1 \) | \(=\) | \(223113289976409774701/32119\) | ||
| \( g_2 \) | \(=\) | \(-681690322946572864/32119\) | ||
| \( g_3 \) | \(=\) | \(2044312783482984/32119\) |
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: \((3 : -19 : 2)\)
magma: [C![3,-19,2]]; // minimal model
magma: [C![3,0,2]]; // 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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(0\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.357951 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 3.221564 \) |
| Analytic order of Ш: | \( 9 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(32119\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 45 T + 32119 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.6.1 | 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);