Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 2x^2 + x$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + x^3z^3 + 2x^2z^4 + xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^6 - 10x^4 + 6x^3 + 9x^2 + 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 1, -3]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 1, -3], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 6, 9, 6, -10, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(1343\) | \(=\) | \( 17 \cdot 79 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(1343\) | \(=\) | \( 17 \cdot 79 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(708\) | \(=\) | \( 2^{2} \cdot 3 \cdot 59 \) |
| \( I_4 \) | \(=\) | \(-32871\) | \(=\) | \( - 3 \cdot 10957 \) |
| \( I_6 \) | \(=\) | \(-7418931\) | \(=\) | \( - 3 \cdot 13^{2} \cdot 14633 \) |
| \( I_{10} \) | \(=\) | \(171904\) | \(=\) | \( 2^{7} \cdot 17 \cdot 79 \) |
| \( J_2 \) | \(=\) | \(177\) | \(=\) | \( 3 \cdot 59 \) |
| \( J_4 \) | \(=\) | \(2675\) | \(=\) | \( 5^{2} \cdot 107 \) |
| \( J_6 \) | \(=\) | \(48537\) | \(=\) | \( 3^{2} \cdot 5393 \) |
| \( J_8 \) | \(=\) | \(358856\) | \(=\) | \( 2^{3} \cdot 31 \cdot 1447 \) |
| \( J_{10} \) | \(=\) | \(1343\) | \(=\) | \( 17 \cdot 79 \) |
| \( g_1 \) | \(=\) | \(173726604657/1343\) | ||
| \( g_2 \) | \(=\) | \(14833498275/1343\) | ||
| \( g_3 \) | \(=\) | \(1520615673/1343\) |
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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) |
| \((1 : 16 : 4)\) | \((1 : -97 : 4)\) | ||||
| All points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) |
| \((1 : 16 : 4)\) | \((1 : -97 : 4)\) | ||||
| All points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) |
| \((1 : -113 : 4)\) | \((1 : 113 : 4)\) | ||||
magma: [C![0,-1,1],C![0,0,1],C![1,-97,4],C![1,-1,0],C![1,0,0],C![1,16,4],C![2,-6,1],C![2,-5,1]]; // minimal model
magma: [C![0,-1,1],C![0,1,1],C![1,-113,4],C![1,-1,0],C![1,1,0],C![1,113,4],C![2,-1,1],C![2,1,1]]; // 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: \(\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.013204\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.013204\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.013204\) | \(\infty\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 0.013204 \) |
| Real period: | \( 18.43111 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.243377 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 17 T^{2} )\) |  |
| \(79\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T + 79 T^{2} )\) | |
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);