Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^5 - 52x^4 + 57x^3 - 9x^2 - 3x - 1$ | (homogenize, simplify) |
$y^2 + z^3y = x^5z - 52x^4z^2 + 57x^3z^3 - 9x^2z^4 - 3xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 208x^4 + 228x^3 - 36x^2 - 12x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -3, -9, 57, -52, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -3, -9, 57, -52, 1], R![1]);
sage: X = HyperellipticCurve(R([-3, -12, -36, 228, -208, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(356211\) | \(=\) | \( 3^{3} \cdot 79 \cdot 167 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-356211\) | \(=\) | \( - 3^{3} \cdot 79 \cdot 167 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(47544\) | \(=\) | \( 2^{3} \cdot 3 \cdot 7 \cdot 283 \) |
\( I_4 \) | \(=\) | \(170813520\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 131 \cdot 1811 \) |
\( I_6 \) | \(=\) | \(1962824156424\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 103 \cdot 191 \cdot 1049 \cdot 1321 \) |
\( I_{10} \) | \(=\) | \(-1424844\) | \(=\) | \( - 2^{2} \cdot 3^{3} \cdot 79 \cdot 167 \) |
\( J_2 \) | \(=\) | \(23772\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 283 \) |
\( J_4 \) | \(=\) | \(-4922754\) | \(=\) | \( - 2 \cdot 3 \cdot 820459 \) |
\( J_6 \) | \(=\) | \(994832028\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 73 \cdot 378551 \) |
\( J_8 \) | \(=\) | \(-146089993725\) | \(=\) | \( - 3^{2} \cdot 5^{2} \cdot 13 \cdot 2383 \cdot 20959 \) |
\( J_{10} \) | \(=\) | \(-356211\) | \(=\) | \( - 3^{3} \cdot 79 \cdot 167 \) |
\( g_1 \) | \(=\) | \(-281167321565803631616/13193\) | ||
\( g_2 \) | \(=\) | \(2449297472511636096/13193\) | ||
\( g_3 \) | \(=\) | \(-20821760065248576/13193\) |
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
All points: \((1 : 0 : 0)\)
magma: [C![1,0,0]]; // minimal model
magma: [C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(1\)
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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
2-torsion field: 5.3.5699376.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 0.488361 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 4.395250 \) |
Analytic order of Ш: | \( 9 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(3\) | \(3\) | \(1\) | \(1 + T\) | |
\(79\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 14 T + 79 T^{2} )\) | |
\(167\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 167 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | 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);