Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^5 - 16x^4 - x^3 + 7x^2 - 3x$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^5z - 16x^4z^2 - x^3z^3 + 7x^2z^4 - 3xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 64x^4 - 4x^3 + 29x^2 - 10x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 7, -1, -16, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 7, -1, -16, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([1, -10, 29, -4, -64, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(100261\) | \(=\) | \( 7 \cdot 14323 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-100261\) | \(=\) | \( - 7 \cdot 14323 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(7048\) | \(=\) | \( 2^{3} \cdot 881 \) |
\( I_4 \) | \(=\) | \(-31736\) | \(=\) | \( - 2^{3} \cdot 3967 \) |
\( I_6 \) | \(=\) | \(-73311541\) | \(=\) | \( - 37 \cdot 1981393 \) |
\( I_{10} \) | \(=\) | \(-401044\) | \(=\) | \( - 2^{2} \cdot 7 \cdot 14323 \) |
\( J_2 \) | \(=\) | \(3524\) | \(=\) | \( 2^{2} \cdot 881 \) |
\( J_4 \) | \(=\) | \(522730\) | \(=\) | \( 2 \cdot 5 \cdot 13 \cdot 4021 \) |
\( J_6 \) | \(=\) | \(104271441\) | \(=\) | \( 3 \cdot 907 \cdot 38321 \) |
\( J_8 \) | \(=\) | \(23551476296\) | \(=\) | \( 2^{3} \cdot 19 \cdot 2017 \cdot 76819 \) |
\( J_{10} \) | \(=\) | \(-100261\) | \(=\) | \( - 7 \cdot 14323 \) |
\( g_1 \) | \(=\) | \(-543474909254042624/100261\) | ||
\( g_2 \) | \(=\) | \(-22876265307259520/100261\) | ||
\( g_3 \) | \(=\) | \(-1294902814688016/100261\) |
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
Known points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -38 : 4),\, (1 : -42 : 4),\, (21 : 972 : 1),\, (21 : -994 : 1)\)
magma: [C![0,-1,1],C![0,0,1],C![1,-42,4],C![1,-38,4],C![1,0,0],C![21,-994,1],C![21,972,1]]; // minimal model
magma: [C![0,-1,1],C![0,1,1],C![1,-4,4],C![1,4,4],C![1,0,0],C![21,-1966,1],C![21,1966,1]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : -42 : 4) - 2 \cdot(1 : 0 : 0)\) | \(x (4x - z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(11xz^2 - 8z^3\) | \(1.274501\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.170227\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : -42 : 4) - 2 \cdot(1 : 0 : 0)\) | \(x (4x - z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(11xz^2 - 8z^3\) | \(1.274501\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.170227\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x (4x - z)\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(23xz^2 - 15z^3\) | \(1.274501\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.170227\) | \(\infty\) |
2-torsion field: 5.3.1604176.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.215449 \) |
Real period: | \( 6.839751 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.473618 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T + 7 T^{2} )\) | |
\(14323\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 140 T + 14323 T^{2} )\) |
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);