Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 + 6x^4 + 10x^3 - 33x^2 - 14x + 3$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 6x^4z^2 + 10x^3z^3 - 33x^2z^4 - 14xz^5 + 3z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 26x^4 + 42x^3 - 131x^2 - 54x + 13$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, -14, -33, 10, 6, 0, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, -14, -33, 10, 6, 0, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([13, -54, -131, 42, 26, 0, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(3732\) | \(=\) | \( 2^{2} \cdot 3 \cdot 311 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(477696\) | \(=\) | \( 2^{9} \cdot 3 \cdot 311 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(37220\) | \(=\) | \( 2^{2} \cdot 5 \cdot 1861 \) |
| \( I_4 \) | \(=\) | \(101230897\) | \(=\) | \( 101230897 \) |
| \( I_6 \) | \(=\) | \(771030827689\) | \(=\) | \( 733807 \cdot 1050727 \) |
| \( I_{10} \) | \(=\) | \(61145088\) | \(=\) | \( 2^{16} \cdot 3 \cdot 311 \) |
| \( J_2 \) | \(=\) | \(9305\) | \(=\) | \( 5 \cdot 1861 \) |
| \( J_4 \) | \(=\) | \(-610328\) | \(=\) | \( - 2^{3} \cdot 23 \cdot 31 \cdot 107 \) |
| \( J_6 \) | \(=\) | \(2058420288\) | \(=\) | \( 2^{6} \cdot 3 \cdot 10720939 \) |
| \( J_8 \) | \(=\) | \(4695275128064\) | \(=\) | \( 2^{8} \cdot 17 \cdot 113 \cdot 2707 \cdot 3527 \) |
| \( J_{10} \) | \(=\) | \(477696\) | \(=\) | \( 2^{9} \cdot 3 \cdot 311 \) |
| \( g_1 \) | \(=\) | \(69756051129891565625/477696\) | ||
| \( g_2 \) | \(=\) | \(-61464229238358875/59712\) | ||
| \( g_3 \) | \(=\) | \(928251329460475/2488\) |
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: | \( 0.383044 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.766089 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(9\) | \(1\) | \(1 - T + T^{2}\) | |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) |  |
| \(311\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 18 T + 311 T^{2} )\) |  |
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);