Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = -2x^6 + x^5 - x^4 + 9x^3 - 2x^2 + 2x - 9$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = -2x^6 + x^5z - x^4z^2 + 9x^3z^3 - 2x^2z^4 + 2xz^5 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -8x^6 + 4x^5 - 4x^4 + 36x^3 - 7x^2 + 10x - 35$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 2, -2, 9, -1, 1, -2]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 2, -2, 9, -1, 1, -2], R![1, 1]);
sage: X = HyperellipticCurve(R([-35, 10, -7, 36, -4, 4, -8]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1870\) | \(=\) | \( 2 \cdot 5 \cdot 11 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(226270\) | \(=\) | \( 2 \cdot 5 \cdot 11^{3} \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(14568\) | \(=\) | \( 2^{3} \cdot 3 \cdot 607 \) |
\( I_4 \) | \(=\) | \(14455716\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 97 \cdot 1129 \) |
\( I_6 \) | \(=\) | \(51876443577\) | \(=\) | \( 3 \cdot 7 \cdot 1439 \cdot 1716683 \) |
\( I_{10} \) | \(=\) | \(-905080\) | \(=\) | \( - 2^{3} \cdot 5 \cdot 11^{3} \cdot 17 \) |
\( J_2 \) | \(=\) | \(7284\) | \(=\) | \( 2^{2} \cdot 3 \cdot 607 \) |
\( J_4 \) | \(=\) | \(-198592\) | \(=\) | \( - 2^{6} \cdot 29 \cdot 107 \) |
\( J_6 \) | \(=\) | \(5333559\) | \(=\) | \( 3 \cdot 7 \cdot 11^{2} \cdot 2099 \) |
\( J_8 \) | \(=\) | \(-147284677\) | \(=\) | \( - 79 \cdot 1864363 \) |
\( J_{10} \) | \(=\) | \(-226270\) | \(=\) | \( - 2 \cdot 5 \cdot 11^{3} \cdot 17 \) |
\( g_1 \) | \(=\) | \(-10252261852576307712/113135\) | ||
\( g_2 \) | \(=\) | \(38374397094057984/113135\) | ||
\( g_3 \) | \(=\) | \(-1169342169912/935\) |
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: \(\Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9x^2 + 10xz + 15z^2\) | \(=\) | \(0,\) | \(81y\) | \(=\) | \(-32xz^2 - 48z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9x^2 + 10xz + 15z^2\) | \(=\) | \(0,\) | \(81y\) | \(=\) | \(-32xz^2 - 48z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(9x^2 + 10xz + 15z^2\) | \(=\) | \(0,\) | \(81y\) | \(=\) | \(-63xz^2 - 95z^3\) | \(0\) | \(3\) |
2-torsion field: 6.2.1674035968000.4
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 1.924846 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 0.641615 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 2 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 5 T^{2} )\) | |
\(11\) | \(1\) | \(3\) | \(3\) | \(( 1 - T )( 1 + 11 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 17 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.20.2 | no |
\(3\) | 3.720.4 | 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);