Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -6x^6 - 19x^5 - 29x^4 - 24x^3 - 8x^2 + 2x + 3$ | (homogenize, simplify) |
$y^2 + xz^2y = -6x^6 - 19x^5z - 29x^4z^2 - 24x^3z^3 - 8x^2z^4 + 2xz^5 + 3z^6$ | (dehomogenize, simplify) |
$y^2 = -24x^6 - 76x^5 - 116x^4 - 96x^3 - 31x^2 + 8x + 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 2, -8, -24, -29, -19, -6]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 2, -8, -24, -29, -19, -6], R![0, 1]);
sage: X = HyperellipticCurve(R([12, 8, -31, -96, -116, -76, -24]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(523925\) | \(=\) | \( 5^{2} \cdot 19 \cdot 1103 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(523925\) | \(=\) | \( 5^{2} \cdot 19 \cdot 1103 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(10640\) | \(=\) | \( 2^{4} \cdot 5 \cdot 7 \cdot 19 \) |
\( I_4 \) | \(=\) | \(1775404\) | \(=\) | \( 2^{2} \cdot 443851 \) |
\( I_6 \) | \(=\) | \(7085263351\) | \(=\) | \( 7085263351 \) |
\( I_{10} \) | \(=\) | \(2095700\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 19 \cdot 1103 \) |
\( J_2 \) | \(=\) | \(5320\) | \(=\) | \( 2^{3} \cdot 5 \cdot 7 \cdot 19 \) |
\( J_4 \) | \(=\) | \(883366\) | \(=\) | \( 2 \cdot 11 \cdot 40153 \) |
\( J_6 \) | \(=\) | \(-1437239\) | \(=\) | \( -1437239 \) |
\( J_8 \) | \(=\) | \(-196995400359\) | \(=\) | \( - 3 \cdot 17 \cdot 47 \cdot 109 \cdot 753983 \) |
\( J_{10} \) | \(=\) | \(523925\) | \(=\) | \( 5^{2} \cdot 19 \cdot 1103 \) |
\( g_1 \) | \(=\) | \(8971489472512000/1103\) | ||
\( g_2 \) | \(=\) | \(280015432238080/1103\) | ||
\( g_3 \) | \(=\) | \(-85636448576/1103\) |
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);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0\) | \(4\) |
2-torsion field: 6.0.16689442262000.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 3.349374 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 3.349374 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 19 T^{2} )\) | |
\(1103\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 48 T + 1103 T^{2} )\) |
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);