Minimal equation
Minimal equation
Simplified equation
| $y^2 + xy = x^5 - 5x^4 + 4x^3 - x$ | (homogenize, simplify) |
| $y^2 + xz^2y = x^5z - 5x^4z^2 + 4x^3z^3 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 - 20x^4 + 16x^3 + x^2 - 4x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 4, -5, 1]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 4, -5, 1], R![0, 1]);
sage: X = HyperellipticCurve(R([0, -4, 1, 16, -20, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(2123\) | \(=\) | \( 11 \cdot 193 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-409739\) | \(=\) | \( - 11 \cdot 193^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(304\) | \(=\) | \( 2^{4} \cdot 19 \) |
| \( I_4 \) | \(=\) | \(8836\) | \(=\) | \( 2^{2} \cdot 47^{2} \) |
| \( I_6 \) | \(=\) | \(508207\) | \(=\) | \( 7 \cdot 79 \cdot 919 \) |
| \( I_{10} \) | \(=\) | \(-1638956\) | \(=\) | \( - 2^{2} \cdot 11 \cdot 193^{2} \) |
| \( J_2 \) | \(=\) | \(152\) | \(=\) | \( 2^{3} \cdot 19 \) |
| \( J_4 \) | \(=\) | \(-510\) | \(=\) | \( - 2 \cdot 3 \cdot 5 \cdot 17 \) |
| \( J_6 \) | \(=\) | \(13841\) | \(=\) | \( 13841 \) |
| \( J_8 \) | \(=\) | \(460933\) | \(=\) | \( 11 \cdot 41903 \) |
| \( J_{10} \) | \(=\) | \(-409739\) | \(=\) | \( - 11 \cdot 193^{2} \) |
| \( g_1 \) | \(=\) | \(-81136812032/409739\) | ||
| \( g_2 \) | \(=\) | \(1791022080/409739\) | ||
| \( g_3 \) | \(=\) | \(-319782464/409739\) |
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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (4 : -2 : 1)\)
magma: [C![0,0,1],C![1,0,0],C![4,-2,1]]; // minimal model
magma: [C![0,0,1],C![1,0,0],C![4,0,1]]; // 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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((4 : -2 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \((4 : -2 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
| \((4 : 0 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 4z^3\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 5.583418 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 0.697927 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 11 T^{2} )\) |  |
| \(193\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 14 T + 193 T^{2} )\) |  |
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);