Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = 14x^6 + 29x^5 + 68x^4 + 71x^3 + 79x^2 + 39x + 22$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = 14x^6 + 29x^5z + 68x^4z^2 + 71x^3z^3 + 79x^2z^4 + 39xz^5 + 22z^6$ | (dehomogenize, simplify) |
$y^2 = 56x^6 + 116x^5 + 273x^4 + 286x^3 + 317x^2 + 156x + 88$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([22, 39, 79, 71, 68, 29, 14]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![22, 39, 79, 71, 68, 29, 14], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([88, 156, 317, 286, 273, 116, 56]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(4667\) | \(=\) | \( 13 \cdot 359 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-4667\) | \(=\) | \( - 13 \cdot 359 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(676380\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 11273 \) |
\( I_4 \) | \(=\) | \(3621945\) | \(=\) | \( 3 \cdot 5 \cdot 241463 \) |
\( I_6 \) | \(=\) | \(814305694467\) | \(=\) | \( 3 \cdot 271435231489 \) |
\( I_{10} \) | \(=\) | \(597376\) | \(=\) | \( 2^{7} \cdot 13 \cdot 359 \) |
\( J_2 \) | \(=\) | \(169095\) | \(=\) | \( 3 \cdot 5 \cdot 11273 \) |
\( J_4 \) | \(=\) | \(1191229045\) | \(=\) | \( 5 \cdot 61 \cdot 3905669 \) |
\( J_6 \) | \(=\) | \(11187800674339\) | \(=\) | \( 41 \cdot 67 \cdot 163 \cdot 811 \cdot 30809 \) |
\( J_8 \) | \(=\) | \(118193629343935295\) | \(=\) | \( 5 \cdot 2423 \cdot 38669 \cdot 252294457 \) |
\( J_{10} \) | \(=\) | \(4667\) | \(=\) | \( 13 \cdot 359 \) |
\( g_1 \) | \(=\) | \(138246399805930957265934375/4667\) | ||
\( g_2 \) | \(=\) | \(5759536994600655307831875/4667\) | ||
\( g_3 \) | \(=\) | \(319894116309350290199475/4667\) |
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);
|
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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(-2xz^2 + 16z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(-2xz^2 + 16z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(x^2z - 3xz^2 + 32z^3\) | \(0\) | \(4\) |
2-torsion field: 6.4.18121699648.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 4.040276 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.010069 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 13 T^{2} )\) | |
\(359\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 8 T + 359 T^{2} )\) |
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);