Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = 4x^5 + 8x^4 + 4x^3$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = 4x^5z + 8x^4z^2 + 4x^3z^3$ | (dehomogenize, simplify) |
$y^2 = 16x^5 + 33x^4 + 18x^3 + 3x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 4, 8, 4]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 4, 8, 4], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([1, 2, 3, 18, 33, 16]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(4276\) | \(=\) | \( 2^{2} \cdot 1069 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(273664\) | \(=\) | \( 2^{8} \cdot 1069 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(820\) | \(=\) | \( 2^{2} \cdot 5 \cdot 41 \) |
\( I_4 \) | \(=\) | \(21577\) | \(=\) | \( 21577 \) |
\( I_6 \) | \(=\) | \(3581701\) | \(=\) | \( 3581701 \) |
\( I_{10} \) | \(=\) | \(35028992\) | \(=\) | \( 2^{15} \cdot 1069 \) |
\( J_2 \) | \(=\) | \(205\) | \(=\) | \( 5 \cdot 41 \) |
\( J_4 \) | \(=\) | \(852\) | \(=\) | \( 2^{2} \cdot 3 \cdot 71 \) |
\( J_6 \) | \(=\) | \(21392\) | \(=\) | \( 2^{4} \cdot 7 \cdot 191 \) |
\( J_8 \) | \(=\) | \(914864\) | \(=\) | \( 2^{4} \cdot 57179 \) |
\( J_{10} \) | \(=\) | \(273664\) | \(=\) | \( 2^{8} \cdot 1069 \) |
\( g_1 \) | \(=\) | \(362050628125/273664\) | ||
\( g_2 \) | \(=\) | \(1835021625/68416\) | ||
\( g_3 \) | \(=\) | \(56187425/17104\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : -2 : 2)\) |
\((-1 : -4 : 2)\) | \((1 : 4 : 2)\) | \((1 : -18 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : -2 : 2)\) |
\((-1 : -4 : 2)\) | \((1 : 4 : 2)\) | \((1 : -18 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((-1 : -2 : 2)\) |
\((-1 : 2 : 2)\) | \((1 : -22 : 2)\) | \((1 : 22 : 2)\) |
magma: [C![-1,-4,2],C![-1,-2,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-18,2],C![1,0,0],C![1,4,2]]; // minimal model
magma: [C![-1,-2,2],C![-1,2,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-22,2],C![1,0,0],C![1,22,2]]; // 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: \(\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 2) + (-1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2 + z^3\) | \(0.003352\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 2) + (-1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2 + z^3\) | \(0.003352\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z + 7xz^2 + 3z^3\) | \(0.003352\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.003352 \) |
Real period: | \( 16.84541 \) |
Tamagawa product: | \( 10 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.564662 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(8\) | \(10\) | \(( 1 - T )( 1 + T )\) | |
\(1069\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 35 T + 1069 T^{2} )\) |
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);