Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2)y = -x^6 + 16x^4 - 112x^2 - 44x + 203$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z)y = -x^6 + 16x^4z^2 - 112x^2z^4 - 44xz^5 + 203z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 2x^5 + 65x^4 - 448x^2 - 176x + 812$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([203, -44, -112, 0, 16, 0, -1]), R([0, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![203, -44, -112, 0, 16, 0, -1], R![0, 0, 1, 1]);
sage: X = HyperellipticCurve(R([812, -176, -448, 0, 65, 2, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(99088\) | \(=\) | \( 2^{4} \cdot 11 \cdot 563 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-198176\) | \(=\) | \( - 2^{5} \cdot 11 \cdot 563 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(259120\) | \(=\) | \( 2^{4} \cdot 5 \cdot 41 \cdot 79 \) |
| \( I_4 \) | \(=\) | \(-110984072\) | \(=\) | \( - 2^{3} \cdot 991 \cdot 13999 \) |
| \( I_6 \) | \(=\) | \(-9341389746260\) | \(=\) | \( - 2^{2} \cdot 5 \cdot 11 \cdot 13^{2} \cdot 5113 \cdot 49139 \) |
| \( I_{10} \) | \(=\) | \(-792704\) | \(=\) | \( - 2^{7} \cdot 11 \cdot 563 \) |
| \( J_2 \) | \(=\) | \(129560\) | \(=\) | \( 2^{3} \cdot 5 \cdot 41 \cdot 79 \) |
| \( J_4 \) | \(=\) | \(717905412\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 2848831 \) |
| \( J_6 \) | \(=\) | \(5406417127940\) | \(=\) | \( 2^{2} \cdot 5 \cdot 151 \cdot 367 \cdot 4877941 \) |
| \( J_8 \) | \(=\) | \(46266805629254164\) | \(=\) | \( 2^{2} \cdot 107 \cdot 2801237 \cdot 38590099 \) |
| \( J_{10} \) | \(=\) | \(-198176\) | \(=\) | \( - 2^{5} \cdot 11 \cdot 563 \) |
| \( g_1 \) | \(=\) | \(-1140787406882816636800000/6193\) | ||
| \( g_2 \) | \(=\) | \(-48789915619039907256000/6193\) | ||
| \( g_3 \) | \(=\) | \(-2835968813284551037000/6193\) |
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
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\Q_{2}$.
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 2xz - 29z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(-97xz^2 - 145z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 2xz - 29z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(-97xz^2 - 145z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(3x^2 - 2xz - 29z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(x^3 + x^2z - 194xz^2 - 290z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.216005498368.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(4\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.887547 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 1.775094 \) |
| Analytic order of Ш: | \( 8 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(5\) | \(1\) | \(1 + T\) | |
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 11 T^{2} )\) |  |
| \(563\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 20 T + 563 T^{2} )\) | |
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);