Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7x^4 + 74x^3 - 145x^2 + 99x - 33$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - x^5z - 7x^4z^2 + 74x^3z^3 - 145x^2z^4 + 99xz^5 - 33z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 4x^5 - 26x^4 + 298x^3 - 579x^2 + 398x - 131$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-33, 99, -145, 74, -7, -1, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-33, 99, -145, 74, -7, -1, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-131, 398, -579, 298, -26, -4, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(961\) | \(=\) | \( 31^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-961\) | \(=\) | \( - 31^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(66980\) | \(=\) | \( 2^{2} \cdot 5 \cdot 17 \cdot 197 \) |
| \( I_4 \) | \(=\) | \(1011437281\) | \(=\) | \( 859 \cdot 1177459 \) |
| \( I_6 \) | \(=\) | \(14016353908561\) | \(=\) | \( 199 \cdot 38281 \cdot 1839919 \) |
| \( I_{10} \) | \(=\) | \(-123008\) | \(=\) | \( - 2^{7} \cdot 31^{2} \) |
| \( J_2 \) | \(=\) | \(16745\) | \(=\) | \( 5 \cdot 17 \cdot 197 \) |
| \( J_4 \) | \(=\) | \(-30460094\) | \(=\) | \( - 2 \cdot 7 \cdot 2175721 \) |
| \( J_6 \) | \(=\) | \(12221475912\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 101 \cdot 560207 \) |
| \( J_8 \) | \(=\) | \(-180792178085599\) | \(=\) | \( - 1447 \cdot 105071 \cdot 1189127 \) |
| \( J_{10} \) | \(=\) | \(-961\) | \(=\) | \( - 31^{2} \) |
| \( g_1 \) | \(=\) | \(-1316514841399349215625/961\) | ||
| \( g_2 \) | \(=\) | \(143016680917998700750/961\) | ||
| \( g_3 \) | \(=\) | \(-3426841043882137800/961\) |
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 $\R$.
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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.224643 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.449287 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(31\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |  |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.15.2 | no |
| \(3\) | 3.72.2 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{5}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);