Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 27x^3 + 26x^2 - 47x - 59$ | (homogenize, simplify) |
| $y^2 + (x^3 + z^3)y = -x^6 - 4x^5z + 27x^3z^3 + 26x^2z^4 - 47xz^5 - 59z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 16x^5 + 110x^3 + 104x^2 - 188x - 235$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-59, -47, 26, 27, 0, -4, -1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-59, -47, 26, 27, 0, -4, -1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-235, -188, 104, 110, 0, -16, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(270438\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 47 \cdot 137 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-270438\) | \(=\) | \( - 2 \cdot 3 \cdot 7 \cdot 47 \cdot 137 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(11860\) | \(=\) | \( 2^{2} \cdot 5 \cdot 593 \) |
| \( I_4 \) | \(=\) | \(9607321\) | \(=\) | \( 227 \cdot 42323 \) |
| \( I_6 \) | \(=\) | \(28052443421\) | \(=\) | \( 151 \cdot 185777771 \) |
| \( I_{10} \) | \(=\) | \(-34616064\) | \(=\) | \( - 2^{8} \cdot 3 \cdot 7 \cdot 47 \cdot 137 \) |
| \( J_2 \) | \(=\) | \(2965\) | \(=\) | \( 5 \cdot 593 \) |
| \( J_4 \) | \(=\) | \(-34004\) | \(=\) | \( - 2^{2} \cdot 8501 \) |
| \( J_6 \) | \(=\) | \(416332\) | \(=\) | \( 2^{2} \cdot 7 \cdot 14869 \) |
| \( J_8 \) | \(=\) | \(19538091\) | \(=\) | \( 3^{4} \cdot 31^{2} \cdot 251 \) |
| \( J_{10} \) | \(=\) | \(-270438\) | \(=\) | \( - 2 \cdot 3 \cdot 7 \cdot 47 \cdot 137 \) |
| \( g_1 \) | \(=\) | \(-229151913706853125/270438\) | ||
| \( g_2 \) | \(=\) | \(443173828089250/135219\) | ||
| \( g_3 \) | \(=\) | \(-261433449050/19317\) |
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: \(\Z/{2}\Z\) (conditional)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 + 2z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 + 2z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 6xz^2 + 5z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.24573935179584.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) (lower bound) |
| 2-Selmer rank: | \(3\) |
| Regulator: | \( 1.0 \) |
| Real period: | \( 1.418182 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 5.672729 \) |
| Analytic order of Ш: | \( 16 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 3 T^{2} )\) |  |
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) |  |
| \(47\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 8 T + 47 T^{2} )\) | |
| \(137\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 18 T + 137 T^{2} )\) | |
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);