Minimal equation
Minimal equation
Simplified equation
| $y^2 + xy = 18x^6 - 35x^5 + 55x^4 - 54x^3 + 34x^2 - 17x + 2$ | (homogenize, simplify) |
| $y^2 + xz^2y = 18x^6 - 35x^5z + 55x^4z^2 - 54x^3z^3 + 34x^2z^4 - 17xz^5 + 2z^6$ | (dehomogenize, simplify) |
| $y^2 = 72x^6 - 140x^5 + 220x^4 - 216x^3 + 137x^2 - 68x + 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -17, 34, -54, 55, -35, 18]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -17, 34, -54, 55, -35, 18], R![0, 1]);
sage: X = HyperellipticCurve(R([8, -68, 137, -216, 220, -140, 72]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(38267\) | \(=\) | \( 17 \cdot 2251 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(38267\) | \(=\) | \( 17 \cdot 2251 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(10064\) | \(=\) | \( 2^{4} \cdot 17 \cdot 37 \) |
| \( I_4 \) | \(=\) | \(90454132\) | \(=\) | \( 2^{2} \cdot 29 \cdot 47^{2} \cdot 353 \) |
| \( I_6 \) | \(=\) | \(590204970799\) | \(=\) | \( 590204970799 \) |
| \( I_{10} \) | \(=\) | \(153068\) | \(=\) | \( 2^{2} \cdot 17 \cdot 2251 \) |
| \( J_2 \) | \(=\) | \(5032\) | \(=\) | \( 2^{3} \cdot 17 \cdot 37 \) |
| \( J_4 \) | \(=\) | \(-14020646\) | \(=\) | \( - 2 \cdot 7010323 \) |
| \( J_6 \) | \(=\) | \(-44210924447\) | \(=\) | \( - 113 \cdot 2693 \cdot 145283 \) |
| \( J_8 \) | \(=\) | \(-104761971518655\) | \(=\) | \( - 3 \cdot 5 \cdot 11 \cdot 181 \cdot 929 \cdot 3775943 \) |
| \( J_{10} \) | \(=\) | \(38267\) | \(=\) | \( 17 \cdot 2251 \) |
| \( g_1 \) | \(=\) | \(189781659896938496/2251\) | ||
| \( g_2 \) | \(=\) | \(-105085065703005184/2251\) | ||
| \( g_3 \) | \(=\) | \(-65850934057921984/2251\) |
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 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: \(\Z/{4}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(2x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(4\) |
2-torsion field: 6.0.6372909033728.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 1 \) |
| Real period: | \( 2.048020 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 2.048020 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(17\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 17 T^{2} )\) |  |
| \(2251\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 2251 T^{2} )\) | |
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);