Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 0, 1], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 0, 1]), R([1]))
$y^2 + y = x^5$
Invariants
| \( N \) | = | \( 3125 \) | = | \( 5^{5} \) | magma: Conductor(LSeries(C)); Factorization($1);
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| \( \Delta \) | = | \(3125\) | = | \( 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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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);
| \( I_2 \) | = | \(0\) | = | \( 0 \) |
| \( I_4 \) | = | \(0\) | = | \( 0 \) |
| \( I_6 \) | = | \(0\) | = | \( 0 \) |
| \( I_{10} \) | = | \(12800000\) | = | \( 2^{12} \cdot 5^{5} \) |
| \( J_2 \) | = | \(0\) | = | \( 0 \) |
| \( J_4 \) | = | \(0\) | = | \( 0 \) |
| \( J_6 \) | = | \(0\) | = | \( 0 \) |
| \( J_8 \) | = | \(0\) | = | \( 0 \) |
| \( J_{10} \) | = | \(3125\) | = | \( 5^{5} \) |
| \( g_1 \) | = | \(0\) | ||
| \( g_2 \) | = | \(0\) | ||
| \( g_3 \) | = | \(0\) |
Automorphism group
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magma: AutomorphismGroup(C); IdentifyGroup($1);
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| \(\mathrm{Aut}(X)\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) | |
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magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(C_{10} \) | (GAP id : [10,2]) | |
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: [C![0,-1,1],C![0,0,1],C![1,0,0]];
All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(1\)
Invariants of the Jacobian:
Analytic rank: \(0\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
2-Selmer rank: \(0\)
magma: HasSquareSha(Jacobian(C));
Order of Ш*: square
Regulator: 1.0
Real period: 17.957840710514119262915086951
Tamagawa numbers: 1 (p = 5)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\Z/{5}\Z\)
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $F_{ac}$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{U}(1)\) |
Decomposition
Simple over \(\overline{\Q}\)
Endomorphisms
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{5})\) with defining polynomial \(x^{4} - x^{3} + x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | the maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q(\zeta_{5})\) (CM) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\C \times \C\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(a^{3} - a^{2}\) with minimal polynomial \(x^{2} - x - 1\):| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
of \(\GL_2\)-type, simple