Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + 3x^2 + x + 2$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5 + 2z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^5 + 7x^4 + 8x^3 + 15x^2 + 6x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 3, 1, 1]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 3, 1, 1], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([9, 6, 15, 8, 7, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(2304\) | \(=\) | \( 2^{8} \cdot 3^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-13824\) | \(=\) | \( - 2^{9} \cdot 3^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(186\) | \(=\) | \( 2 \cdot 3 \cdot 31 \) |
| \( I_4 \) | \(=\) | \(54\) | \(=\) | \( 2 \cdot 3^{3} \) |
| \( I_6 \) | \(=\) | \(2664\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 37 \) |
| \( I_{10} \) | \(=\) | \(54\) | \(=\) | \( 2 \cdot 3^{3} \) |
| \( J_2 \) | \(=\) | \(372\) | \(=\) | \( 2^{2} \cdot 3 \cdot 31 \) |
| \( J_4 \) | \(=\) | \(5622\) | \(=\) | \( 2 \cdot 3 \cdot 937 \) |
| \( J_6 \) | \(=\) | \(115100\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 1151 \) |
| \( J_8 \) | \(=\) | \(2802579\) | \(=\) | \( 3 \cdot 13 \cdot 71861 \) |
| \( J_{10} \) | \(=\) | \(13824\) | \(=\) | \( 2^{9} \cdot 3^{3} \) |
| \( g_1 \) | \(=\) | \(515324718\) | ||
| \( g_2 \) | \(=\) | \(83742501/4\) | ||
| \( g_3 \) | \(=\) | \(27652775/24\) |
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 1 : 1),\, (0 : -2 : 1),\, (-1 : -2 : 1),\, (-1 : 2 : 1)\)
magma: [C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-1,-4,1],C![-1,4,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0]]; // simplified model
Number of rational Weierstrass points: \(0\)
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.099203\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(2\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.099203\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(2\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0.099203\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + 3xz^2 + 3z^3\) | \(0\) | \(2\) |
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\zeta_{24})\)
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 0.099203 \) |
| Real period: | \( 11.87246 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 0.294447 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(8\) | \(9\) | \(2\) | \(1 + 2 T + 2 T^{2}\) | |
| \(3\) | \(2\) | \(3\) | \(2\) | \(1 + 2 T + 3 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.90.6 | 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);