Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, -1, 0, 1], R![1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, -1, 0, 1]), R([1, 1]))
$y^2 + (x + 1)y = x^6 - x^4 + 2x^2 + x$
Invariants
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magma: Conductor(LSeries(C)); Factorization($1);
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| \( N \) | = | \( 15256 \) | = | \( 2^{3} \cdot 1907 \) | |
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magma: Discriminant(C); Factorization(Integers()!$1);
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| \( \Delta \) | = | \(122048\) | = | \( 2^{6} \cdot 1907 \) | |
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);
| \( I_2 \) | = | \(-384\) | = | \( -1 \cdot 2^{7} \cdot 3 \) |
| \( I_4 \) | = | \(67776\) | = | \( 2^{6} \cdot 3 \cdot 353 \) |
| \( I_6 \) | = | \(-13663488\) | = | \( -1 \cdot 2^{8} \cdot 3 \cdot 17791 \) |
| \( I_{10} \) | = | \(499908608\) | = | \( 2^{18} \cdot 1907 \) |
| \( J_2 \) | = | \(-48\) | = | \( -1 \cdot 2^{4} \cdot 3 \) |
| \( J_4 \) | = | \(-610\) | = | \( -1 \cdot 2 \cdot 5 \cdot 61 \) |
| \( J_6 \) | = | \(14052\) | = | \( 2^{2} \cdot 3 \cdot 1171 \) |
| \( J_8 \) | = | \(-261649\) | = | \( -1 \cdot 19 \cdot 47 \cdot 293 \) |
| \( J_{10} \) | = | \(122048\) | = | \( 2^{6} \cdot 1907 \) |
| \( g_1 \) | = | \(-3981312/1907\) | ||
| \( g_2 \) | = | \(1054080/1907\) | ||
| \( g_3 \) | = | \(505872/1907\) |
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,-31,4],C![-1,-17,4],C![-1,-3,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-62,3],C![2,17,3]];
Known rational points: (-1 : -31 : 4), (-1 : -17 : 4), (-1 : -3 : 2), (-1 : -1 : 1), (-1 : -1 : 2), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -3 : 1), (1 : -1 : 0), (1 : 1 : 0), (1 : 1 : 1), (2 : -62 : 3), (2 : 17 : 3)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank*: \(2\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
2-Selmer rank: \(2\)
magma: HasSquareSha(Jacobian(C));
Order of Ш*: square
Regulator: 0.00566345688687
Real period: 17.793228555670985757104297985
Tamagawa numbers: 6 (p = 2), 1 (p = 1907)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\mathrm{trivial}\)
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 \(\Q\).