Minimal equation
Minimal equation
Simplified equation
| $y^2 + x^3y = 6x^3 + 11x^2 + 6x + 1$ | (homogenize, simplify) |
| $y^2 + x^3y = 6x^3z^3 + 11x^2z^4 + 6xz^5 + z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 24x^3 + 44x^2 + 24x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 6, 11, 6]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 6, 11, 6], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([4, 24, 44, 24, 0, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(120436\) | \(=\) | \( 2^{2} \cdot 30109 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-481744\) | \(=\) | \( - 2^{4} \cdot 30109 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(312\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \) |
| \( I_4 \) | \(=\) | \(4677\) | \(=\) | \( 3 \cdot 1559 \) |
| \( I_6 \) | \(=\) | \(368463\) | \(=\) | \( 3 \cdot 263 \cdot 467 \) |
| \( I_{10} \) | \(=\) | \(-60218\) | \(=\) | \( - 2 \cdot 30109 \) |
| \( J_2 \) | \(=\) | \(312\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \) |
| \( J_4 \) | \(=\) | \(938\) | \(=\) | \( 2 \cdot 7 \cdot 67 \) |
| \( J_6 \) | \(=\) | \(13008\) | \(=\) | \( 2^{4} \cdot 3 \cdot 271 \) |
| \( J_8 \) | \(=\) | \(794663\) | \(=\) | \( 139 \cdot 5717 \) |
| \( J_{10} \) | \(=\) | \(-481744\) | \(=\) | \( - 2^{4} \cdot 30109 \) |
| \( g_1 \) | \(=\) | \(-184779159552/30109\) | ||
| \( g_2 \) | \(=\) | \(-1780519104/30109\) | ||
| \( g_3 \) | \(=\) | \(-79140672/30109\) |
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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) |
| \((-1 : 0 : 2)\) | \((-1 : 1 : 2)\) | \((-1 : 0 : 3)\) | \((-1 : 1 : 3)\) | \((-2 : 3 : 1)\) | \((-2 : 5 : 1)\) |
| \((2 : 7 : 1)\) | \((3 : 8 : 1)\) | \((2 : -15 : 1)\) | \((3 : -35 : 1)\) | \((-6 : 91 : 5)\) | \((-6 : 125 : 5)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : 1 : 1)\) |
| \((-1 : 0 : 2)\) | \((-1 : 1 : 2)\) | \((-1 : 0 : 3)\) | \((-1 : 1 : 3)\) | \((-2 : 3 : 1)\) | \((-2 : 5 : 1)\) |
| \((2 : 7 : 1)\) | \((3 : 8 : 1)\) | \((2 : -15 : 1)\) | \((3 : -35 : 1)\) | \((-6 : 91 : 5)\) | \((-6 : 125 : 5)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
| \((-1 : -1 : 2)\) | \((-1 : 1 : 2)\) | \((-2 : -2 : 1)\) | \((-2 : 2 : 1)\) | \((-1 : -1 : 3)\) | \((-1 : 1 : 3)\) |
| \((2 : -22 : 1)\) | \((2 : 22 : 1)\) | \((-6 : -34 : 5)\) | \((-6 : 34 : 5)\) | \((3 : -43 : 1)\) | \((3 : 43 : 1)\) |
magma: [C![-6,91,5],C![-6,125,5],C![-2,3,1],C![-2,5,1],C![-1,0,1],C![-1,0,2],C![-1,0,3],C![-1,1,1],C![-1,1,2],C![-1,1,3],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![2,-15,1],C![2,7,1],C![3,-35,1],C![3,8,1]]; // minimal model
magma: [C![-6,-34,5],C![-6,34,5],C![-2,-2,1],C![-2,2,1],C![-1,-1,1],C![-1,-1,2],C![-1,-1,3],C![-1,1,1],C![-1,1,2],C![-1,1,3],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,1,0],C![2,-22,1],C![2,22,1],C![3,-43,1],C![3,43,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 1 : 2) - (1 : 0 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.678297\) | \(\infty\) |
| \((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.392597\) | \(\infty\) |
| \((-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.069829\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 1 : 2) - (1 : 0 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.678297\) | \(\infty\) |
| \((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.392597\) | \(\infty\) |
| \((-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.069829\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 1 : 2) - (1 : 1 : 0)\) | \(z (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.678297\) | \(\infty\) |
| \((-1 : -1 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.392597\) | \(\infty\) |
| \((-1 : -1 : 1) + (0 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 2xz^2 + 2z^3\) | \(0.069829\) | \(\infty\) |
2-torsion field: 6.4.7707904.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(3\) (upper bound) |
| Mordell-Weil rank: | \(3\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 0.018135 \) |
| Real period: | \( 17.82099 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.969564 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(4\) | \(3\) | \(1 + 2 T + 2 T^{2}\) | |
| \(30109\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 200 T + 30109 T^{2} )\) |  |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);