Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + x)y = x^4 + 3x^3 + 3x^2 + 6x + 1$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + xz^2)y = x^4z^2 + 3x^3z^3 + 3x^2z^4 + 6xz^5 + z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^5 + 7x^4 + 14x^3 + 13x^2 + 24x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 6, 3, 3, 1]), R([0, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 6, 3, 3, 1], R![0, 1, 1, 1]);
sage: X = HyperellipticCurve(R([4, 24, 13, 14, 7, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(2173\) | \(=\) | \( 41 \cdot 53 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(89093\) | \(=\) | \( 41^{2} \cdot 53 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(340\) | \(=\) | \( 2^{2} \cdot 5 \cdot 17 \) |
| \( I_4 \) | \(=\) | \(114841\) | \(=\) | \( 41 \cdot 2801 \) |
| \( I_6 \) | \(=\) | \(25395389\) | \(=\) | \( 25395389 \) |
| \( I_{10} \) | \(=\) | \(11403904\) | \(=\) | \( 2^{7} \cdot 41^{2} \cdot 53 \) |
| \( J_2 \) | \(=\) | \(85\) | \(=\) | \( 5 \cdot 17 \) |
| \( J_4 \) | \(=\) | \(-4484\) | \(=\) | \( - 2^{2} \cdot 19 \cdot 59 \) |
| \( J_6 \) | \(=\) | \(-238312\) | \(=\) | \( - 2^{3} \cdot 29789 \) |
| \( J_8 \) | \(=\) | \(-10090694\) | \(=\) | \( - 2 \cdot 5045347 \) |
| \( J_{10} \) | \(=\) | \(89093\) | \(=\) | \( 41^{2} \cdot 53 \) |
| \( g_1 \) | \(=\) | \(4437053125/89093\) | ||
| \( g_2 \) | \(=\) | \(-2753736500/89093\) | ||
| \( g_3 \) | \(=\) | \(-1721804200/89093\) |
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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1)\)
magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![0,-2,1],C![0,2,1],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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.072534\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2 + 4z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.072534\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2 + 4z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2\) | \(0.072534\) | \(\infty\) |
| \(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 + x^2z + 7xz^2 + 8z^3\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.072534 \) |
| Real period: | \( 8.214958 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 0.297932 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(41\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 41 T^{2} )\) |  |
| \(53\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 53 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.15.1 | 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);