Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-36, -98, -41, 48, 9, -4, -1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-36, -98, -41, 48, 9, -4, -1]), R([1, 0, 0, 1]))
$y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 9x^4 + 48x^3 - 41x^2 - 98x - 36$
Invariants
magma: Conductor(LSeries(C)); Factorization($1);
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\( N \) | = | \( 708 \) | = | \( 2^{2} \cdot 3 \cdot 59 \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(-181248\) | = | \( -1 \cdot 2^{10} \cdot 3 \cdot 59 \) |
Igusa-Clebsch invariants
magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Igusa invariants
magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
G2 invariants
magma: G2Invariants(C);
\( I_2 \) | = | \(468200\) | = | \( 2^{3} \cdot 5^{2} \cdot 2341 \) |
\( I_4 \) | = | \(13875516100\) | = | \( 2^{2} \cdot 5^{2} \cdot 138755161 \) |
\( I_6 \) | = | \(1620683728649416\) | = | \( 2^{3} \cdot 1069463 \cdot 189427279 \) |
\( I_{10} \) | = | \(-742391808\) | = | \( -1 \cdot 2^{22} \cdot 3 \cdot 59 \) |
\( J_2 \) | = | \(58525\) | = | \( 5^{2} \cdot 2341 \) |
\( J_4 \) | = | \(-1820975\) | = | \( -1 \cdot 5^{2} \cdot 13^{2} \cdot 431 \) |
\( J_6 \) | = | \(60952909\) | = | \( 109 \cdot 559201 \) |
\( J_8 \) | = | \(62829762150\) | = | \( 2 \cdot 3 \cdot 5^{2} \cdot 47 \cdot 8912023 \) |
\( J_{10} \) | = | \(-181248\) | = | \( -1 \cdot 2^{10} \cdot 3 \cdot 59 \) |
\( g_1 \) | = | \(-686605237334059580078125/181248\) | ||
\( g_2 \) | = | \(365029741228054296875/181248\) | ||
\( g_3 \) | = | \(-208774418179643125/181248\) |
Automorphism group
magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X)\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) | ||
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) |
Rational points
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);
This curve is locally solvable everywhere.
magma: [];
There are no rational points.
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank: \(0\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
2-Selmer rank: \(3\)
magma: HasSquareSha(Jacobian(C));
Order of Ш*: square
Regulator: 1.0
Real period: 0.32534362630909308619897744650
Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 59)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\Z/{2}\Z\)
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition
Simple over \(\overline{\Q}\)
Endomorphisms
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 \(\Q\).