Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, 0, 0, 1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, 0, 0, 1]), R([1, 0, 0, 1]))
$y^2 + (x^3 + 1)y = x^6 + 2x^3 - x$
Invariants
| \( N \) | = | \( 12500 \) | = | \( 2^{2} \cdot 5^{5} \) | magma: Conductor(LSeries(C)); Factorization($1);
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| \( \Delta \) | = | \(12500\) | = | \( 2^{2} \cdot 5^{5} \) | 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);
| \( I_2 \) | = | \(-600\) | = | \( -1 \cdot 2^{3} \cdot 3 \cdot 5^{2} \) |
| \( I_4 \) | = | \(202500\) | = | \( 2^{2} \cdot 3^{4} \cdot 5^{4} \) |
| \( I_6 \) | = | \(-67995000\) | = | \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{4} \cdot 1511 \) |
| \( I_{10} \) | = | \(51200000\) | = | \( 2^{14} \cdot 5^{5} \) |
| \( J_2 \) | = | \(-75\) | = | \( -1 \cdot 3 \cdot 5^{2} \) |
| \( J_4 \) | = | \(-1875\) | = | \( -1 \cdot 3 \cdot 5^{4} \) |
| \( J_6 \) | = | \(73125\) | = | \( 3^{2} \cdot 5^{4} \cdot 13 \) |
| \( J_8 \) | = | \(-2250000\) | = | \( -1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{6} \) |
| \( J_{10} \) | = | \(12500\) | = | \( 2^{2} \cdot 5^{5} \) |
| \( g_1 \) | = | \(-759375/4\) | ||
| \( g_2 \) | = | \(253125/4\) | ||
| \( g_3 \) | = | \(131625/4\) |
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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| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(C_2 \) | (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: [C![-1,0,1],C![0,-1,1],C![0,0,1]];
All rational points: (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1)
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: 25.011388058069645358352525311
Tamagawa numbers: 1 (p = 2), 1 (p = 5)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\Z/{5}\Z\)
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $N(G_{3,3})$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)
Of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |