Minimal equation
Minimal equation
Simplified equation
| $y^2 = 2x^5 - 4x^3 - x^2 + 2x + 1$ | (homogenize, simplify) |
| $y^2 = 2x^5z - 4x^3z^3 - x^2z^4 + 2xz^5 + z^6$ | (dehomogenize, simplify) |
| $y^2 = 2x^5 - 4x^3 - x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, -1, -4, 0, 2]), R([]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 2, -1, -4, 0, 2], R![]);
sage: X = HyperellipticCurve(R([1, 2, -1, -4, 0, 2]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(1408\) | \(=\) | \( 2^{7} \cdot 11 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1408,2),R![1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-180224\) | \(=\) | \( - 2^{14} \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(128\) | \(=\) | \( 2^{7} \) |
| \( I_4 \) | \(=\) | \(-20\) | \(=\) | \( - 2^{2} \cdot 5 \) |
| \( I_6 \) | \(=\) | \(-2290\) | \(=\) | \( - 2 \cdot 5 \cdot 229 \) |
| \( I_{10} \) | \(=\) | \(-22\) | \(=\) | \( - 2 \cdot 11 \) |
| \( J_2 \) | \(=\) | \(512\) | \(=\) | \( 2^{9} \) |
| \( J_4 \) | \(=\) | \(11136\) | \(=\) | \( 2^{7} \cdot 3 \cdot 29 \) |
| \( J_6 \) | \(=\) | \(410624\) | \(=\) | \( 2^{10} \cdot 401 \) |
| \( J_8 \) | \(=\) | \(21557248\) | \(=\) | \( 2^{12} \cdot 19 \cdot 277 \) |
| \( J_{10} \) | \(=\) | \(-180224\) | \(=\) | \( - 2^{14} \cdot 11 \) |
| \( g_1 \) | \(=\) | \(-2147483648/11\) | ||
| \( g_2 \) | \(=\) | \(-91226112/11\) | ||
| \( g_3 \) | \(=\) | \(-6569984/11\) |
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: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1)\)
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![-1,0,1],C![0,-1/2,1],C![0,1/2,1],C![1,0,0],C![1,0,1]]; // simplified model
Number of rational Weierstrass points: \(3\)
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: \(\Z/{2}\Z \oplus \Z/{8}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((-1 : 0 : 1) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(8\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((-1 : 0 : 1) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(8\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((-1 : 0 : 1) + (0 : 1/2 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2xz^2 + 1/2z^3\) | \(0\) | \(8\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 15.31272 \) |
| Tamagawa product: | \( 8 \) |
| Torsion order: | \( 16 \) |
| Leading coefficient: | \( 0.478522 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(7\) | \(14\) | \(8\) | \(1\) | |
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 11 T^{2} )\) |  |
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? |
|---|---|---|
| \(2\) | 2.120.3 | 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);