Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x^2 + 1)y = -x^6 + 2x^5 - 33x^3 + 43x^2 + 15x - 330$ | (homogenize, simplify) |
| $y^2 + (x^3 + x^2z + z^3)y = -x^6 + 2x^5z - 33x^3z^3 + 43x^2z^4 + 15xz^5 - 330z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 + 10x^5 + x^4 - 130x^3 + 174x^2 + 60x - 1319$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-330, 15, 43, -33, 0, 2, -1]), R([1, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-330, 15, 43, -33, 0, 2, -1], R![1, 0, 1, 1]);
sage: X = HyperellipticCurve(R([-1319, 60, 174, -130, 1, 10, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(1253\) | \(=\) | \( 7 \cdot 179 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-1253\) | \(=\) | \( - 7 \cdot 179 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(413532\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 7 \cdot 547 \) |
| \( I_4 \) | \(=\) | \(9381037161\) | \(=\) | \( 3 \cdot 3127012387 \) |
| \( I_6 \) | \(=\) | \(999361725629499\) | \(=\) | \( 3^{3} \cdot 739 \cdot 24019 \cdot 2085257 \) |
| \( I_{10} \) | \(=\) | \(160384\) | \(=\) | \( 2^{7} \cdot 7 \cdot 179 \) |
| \( J_2 \) | \(=\) | \(103383\) | \(=\) | \( 3^{3} \cdot 7 \cdot 547 \) |
| \( J_4 \) | \(=\) | \(54458647\) | \(=\) | \( 54458647 \) |
| \( J_6 \) | \(=\) | \(-97243994481\) | \(=\) | \( - 3 \cdot 29 \cdot 79 \cdot 179 \cdot 79043 \) |
| \( J_8 \) | \(=\) | \(-3254780028624958\) | \(=\) | \( - 2 \cdot 47 \cdot 34625319453457 \) |
| \( J_{10} \) | \(=\) | \(1253\) | \(=\) | \( 7 \cdot 179 \) |
| \( g_1 \) | \(=\) | \(1687126365978608485162449/179\) | ||
| \( g_2 \) | \(=\) | \(8596391751971448839127/179\) | ||
| \( g_3 \) | \(=\) | \(-829487756384515053\) |
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: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 0.207463 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.414927 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) |  |
| \(179\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 179 T^{2} )\) | |
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? |
|---|---|---|
| \(3\) | 3.720.5 | no |
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);