Minimal equation
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -1, -1, 1], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -1, -1, 1]), R([0, 1, 0, 1]))
$y^2 + (x^3 + x)y = x^3 - x^2 - x + 1$
Invariants
|
magma: Conductor(LSeries(C)); Factorization($1);
|
|||||
| \( N \) | = | \( 13016 \) | = | \( 2^{3} \cdot 1627 \) | |
|
magma: Discriminant(C); Factorization(Integers()!$1);
|
|||||
| \( \Delta \) | = | \(-104128\) | = | \( -1 \cdot 2^{6} \cdot 1627 \) | |
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);
| \( I_2 \) | = | \(-768\) | = | \( -1 \cdot 2^{8} \cdot 3 \) |
| \( I_4 \) | = | \(60864\) | = | \( 2^{6} \cdot 3 \cdot 317 \) |
| \( I_6 \) | = | \(-14035200\) | = | \( -1 \cdot 2^{8} \cdot 3 \cdot 5^{2} \cdot 17 \cdot 43 \) |
| \( I_{10} \) | = | \(-426508288\) | = | \( -1 \cdot 2^{18} \cdot 1627 \) |
| \( J_2 \) | = | \(-96\) | = | \( -1 \cdot 2^{5} \cdot 3 \) |
| \( J_4 \) | = | \(-250\) | = | \( -1 \cdot 2 \cdot 5^{3} \) |
| \( J_6 \) | = | \(5412\) | = | \( 2^{2} \cdot 3 \cdot 11 \cdot 41 \) |
| \( J_8 \) | = | \(-145513\) | = | \( -1 \cdot 145513 \) |
| \( J_{10} \) | = | \(-104128\) | = | \( -1 \cdot 2^{6} \cdot 1627 \) |
| \( g_1 \) | = | \(127401984/1627\) | ||
| \( g_2 \) | = | \(-3456000/1627\) | ||
| \( g_3 \) | = | \(-779328/1627\) |
Automorphism group
|
magma: AutomorphismGroup(C); IdentifyGroup($1);
|
|||||
| \(\mathrm{Aut}(X)\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) | ||
|
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
|||||
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(C_2 \) | (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![-2,1,1],C![-2,9,1],C![-1,0,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-8,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,3,2],C![3,-40,2],C![3,1,2]];
Known rational points: (-2 : 1 : 1), (-2 : 9 : 1), (-1 : 0 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -8 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (1 : 3 : 2), (3 : -40 : 2), (3 : 1 : 2)
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: 0.00560889537685
Real period: 16.719158368499771485725916451
Tamagawa numbers: 6 (p = 2), 1 (p = 1627)
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 \(\Q\).