Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -11x^6 - 9x^5 + 39x^4 - 47x^2 + 32x - 7$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -11x^6 - 9x^5z + 39x^4z^2 - 47x^2z^4 + 32xz^5 - 7z^6$ | (dehomogenize, simplify) |
$y^2 = -43x^6 - 36x^5 + 158x^4 + 2x^3 - 187x^2 + 130x - 27$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, 32, -47, 0, 39, -9, -11]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-7, 32, -47, 0, 39, -9, -11], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-27, 130, -187, 2, 158, -36, -43]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(516443\) | \(=\) | \( 17^{2} \cdot 1787 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-516443\) | \(=\) | \( - 17^{2} \cdot 1787 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3460\) | \(=\) | \( 2^{2} \cdot 5 \cdot 173 \) |
\( I_4 \) | \(=\) | \(30814345\) | \(=\) | \( 5 \cdot 79 \cdot 181 \cdot 431 \) |
\( I_6 \) | \(=\) | \(-18826218827\) | \(=\) | \( - 13 \cdot 33587 \cdot 43117 \) |
\( I_{10} \) | \(=\) | \(-66104704\) | \(=\) | \( - 2^{7} \cdot 17^{2} \cdot 1787 \) |
\( J_2 \) | \(=\) | \(865\) | \(=\) | \( 5 \cdot 173 \) |
\( J_4 \) | \(=\) | \(-1252755\) | \(=\) | \( - 3^{2} \cdot 5 \cdot 7 \cdot 41 \cdot 97 \) |
\( J_6 \) | \(=\) | \(571473541\) | \(=\) | \( 613 \cdot 932257 \) |
\( J_8 \) | \(=\) | \(-268767619265\) | \(=\) | \( - 5 \cdot 53753523853 \) |
\( J_{10} \) | \(=\) | \(-516443\) | \(=\) | \( - 17^{2} \cdot 1787 \) |
\( g_1 \) | \(=\) | \(-484262162790625/516443\) | ||
\( g_2 \) | \(=\) | \(810801357541875/516443\) | ||
\( g_3 \) | \(=\) | \(-427590790214725/516443\) |
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 except over $\R$ and $\Q_{2}$.
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));
2-torsion field: 6.0.33052352.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 0.531605 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 9.568904 \) |
Analytic order of Ш: | \( 18 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(17\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(1787\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 13 T + 1787 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);