Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = 2x^5 + 6x^4 - 16x^3 + 8x^2 + x - 1$ | (homogenize, simplify) |
$y^2 + xz^2y = 2x^5z + 6x^4z^2 - 16x^3z^3 + 8x^2z^4 + xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = 8x^5 + 24x^4 - 64x^3 + 33x^2 + 4x - 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 8, -16, 6, 2]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 8, -16, 6, 2], R![0, 1]);
sage: X = HyperellipticCurve(R([-4, 4, 33, -64, 24, 8]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2520\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(635040\) | \(=\) | \( 2^{5} \cdot 3^{4} \cdot 5 \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3296\) | \(=\) | \( 2^{5} \cdot 103 \) |
\( I_4 \) | \(=\) | \(104464\) | \(=\) | \( 2^{4} \cdot 6529 \) |
\( I_6 \) | \(=\) | \(104578588\) | \(=\) | \( 2^{2} \cdot 3467 \cdot 7541 \) |
\( I_{10} \) | \(=\) | \(2540160\) | \(=\) | \( 2^{7} \cdot 3^{4} \cdot 5 \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(1648\) | \(=\) | \( 2^{4} \cdot 103 \) |
\( J_4 \) | \(=\) | \(95752\) | \(=\) | \( 2^{3} \cdot 11969 \) |
\( J_6 \) | \(=\) | \(6711044\) | \(=\) | \( 2^{2} \cdot 41 \cdot 151 \cdot 271 \) |
\( J_8 \) | \(=\) | \(472838752\) | \(=\) | \( 2^{5} \cdot 14776211 \) |
\( J_{10} \) | \(=\) | \(635040\) | \(=\) | \( 2^{5} \cdot 3^{4} \cdot 5 \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(379870928666624/19845\) | ||
\( g_2 \) | \(=\) | \(13392741850112/19845\) | ||
\( g_3 \) | \(=\) | \(569579726368/19845\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -2 : 2)\)
magma: [C![1,-2,2],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![1,0,2],C![1,-1,1],C![1,0,0],C![1,1,1]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z/{2}\Z \oplus \Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(8x^2 - xz - 5z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(-41xz^2 + 27z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(8x^2 - xz - 5z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(-41xz^2 + 27z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 - 2xz - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(8x^2 - xz - 5z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(-81xz^2 + 54z^3\) | \(0\) | \(10\) |
2-torsion field: \(\Q(\sqrt{2}, \sqrt{5})\)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 16.30622 \) |
Tamagawa product: | \( 20 \) |
Torsion order: | \( 20 \) |
Leading coefficient: | \( 0.815311 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(5\) | \(5\) | \(1 - T\) | |
\(3\) | \(2\) | \(4\) | \(2\) | \(1 + T^{2}\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 2 T + 7 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | yes |
\(5\) | not computed | 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);