Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = 2x^5 - 9x^4 + 9x^3 - 2x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = 2x^5z - 9x^4z^2 + 9x^3z^3 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = 8x^5 - 35x^4 + 38x^3 + 3x^2 - 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 0, 9, -9, 2]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 0, 9, -9, 2], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([1, -6, 3, 38, -35, 8]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(249362\) | \(=\) | \( 2 \cdot 41 \cdot 3041 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-498724\) | \(=\) | \( - 2^{2} \cdot 41 \cdot 3041 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(4212\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 13 \) |
\( I_4 \) | \(=\) | \(154569\) | \(=\) | \( 3 \cdot 67 \cdot 769 \) |
\( I_6 \) | \(=\) | \(192076389\) | \(=\) | \( 3^{2} \cdot 21341821 \) |
\( I_{10} \) | \(=\) | \(-63836672\) | \(=\) | \( - 2^{9} \cdot 41 \cdot 3041 \) |
\( J_2 \) | \(=\) | \(1053\) | \(=\) | \( 3^{4} \cdot 13 \) |
\( J_4 \) | \(=\) | \(39760\) | \(=\) | \( 2^{4} \cdot 5 \cdot 7 \cdot 71 \) |
\( J_6 \) | \(=\) | \(1918804\) | \(=\) | \( 2^{2} \cdot 479701 \) |
\( J_8 \) | \(=\) | \(109910753\) | \(=\) | \( 67 \cdot 127 \cdot 12917 \) |
\( J_{10} \) | \(=\) | \(-498724\) | \(=\) | \( - 2^{2} \cdot 41 \cdot 3041 \) |
\( g_1 \) | \(=\) | \(-1294618640600493/498724\) | ||
\( g_2 \) | \(=\) | \(-11605704217380/124681\) | ||
\( g_3 \) | \(=\) | \(-531896786109/124681\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 2)\) | \((1 : -3 : 1)\) |
\((2 : -3 : 1)\) | \((2 : -4 : 1)\) | \((3 : -6 : 2)\) | \((1 : -12 : 2)\) | \((5 : 29 : 1)\) | \((3 : -32 : 2)\) |
\((5 : -60 : 1)\) | \((7 : -2184 : 18)\) | \((7 : -6798 : 18)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -2 : 2)\) | \((1 : -3 : 1)\) |
\((2 : -3 : 1)\) | \((2 : -4 : 1)\) | \((3 : -6 : 2)\) | \((1 : -12 : 2)\) | \((5 : 29 : 1)\) | \((3 : -32 : 2)\) |
\((5 : -60 : 1)\) | \((7 : -2184 : 18)\) | \((7 : -6798 : 18)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((1 : -3 : 1)\) |
\((1 : 3 : 1)\) | \((1 : -10 : 2)\) | \((1 : 10 : 2)\) | \((3 : -26 : 2)\) | \((3 : 26 : 2)\) | \((5 : -89 : 1)\) |
\((5 : 89 : 1)\) | \((7 : -4614 : 18)\) | \((7 : 4614 : 18)\) |
magma: [C![0,-1,1],C![0,0,1],C![1,-12,2],C![1,-3,1],C![1,-2,2],C![1,0,0],C![1,0,1],C![2,-4,1],C![2,-3,1],C![3,-32,2],C![3,-6,2],C![5,-60,1],C![5,29,1],C![7,-6798,18],C![7,-2184,18]]; // minimal model
magma: [C![0,-1,1],C![0,1,1],C![1,-10,2],C![1,-3,1],C![1,10,2],C![1,0,0],C![1,3,1],C![2,-1,1],C![2,1,1],C![3,-26,2],C![3,26,2],C![5,-89,1],C![5,89,1],C![7,-4614,18],C![7,4614,18]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.880650\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.293429\) | \(\infty\) |
\((0 : -1 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.204690\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.880650\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.293429\) | \(\infty\) |
\((0 : -1 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.204690\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 3xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2 + z^3\) | \(0.880650\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 - z^3\) | \(0.293429\) | \(\infty\) |
\((0 : -1 : 1) + (2 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2 - z^3\) | \(0.204690\) | \(\infty\) |
2-torsion field: 5.3.1994896.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.044768 \) |
Real period: | \( 16.47752 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.475362 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(41\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 10 T + 41 T^{2} )\) | |
\(3041\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 46 T + 3041 T^{2} )\) |
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);