Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -2x^4 - x^3 + x + 1$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -2x^4z^2 - x^3z^3 + xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 6x^4 - 4x^3 + x^2 + 4x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 1, 0, -1, -2]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 1, 0, -1, -2], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([4, 4, 1, -4, -6, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1416\) | \(=\) | \( 2^{3} \cdot 3 \cdot 59 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-135936\) | \(=\) | \( - 2^{8} \cdot 3^{2} \cdot 59 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(192\) | \(=\) | \( 2^{6} \cdot 3 \) |
\( I_4 \) | \(=\) | \(-96\) | \(=\) | \( - 2^{5} \cdot 3 \) |
\( I_6 \) | \(=\) | \(90660\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 1511 \) |
\( I_{10} \) | \(=\) | \(543744\) | \(=\) | \( 2^{10} \cdot 3^{2} \cdot 59 \) |
\( J_2 \) | \(=\) | \(96\) | \(=\) | \( 2^{5} \cdot 3 \) |
\( J_4 \) | \(=\) | \(400\) | \(=\) | \( 2^{4} \cdot 5^{2} \) |
\( J_6 \) | \(=\) | \(-8452\) | \(=\) | \( - 2^{2} \cdot 2113 \) |
\( J_8 \) | \(=\) | \(-242848\) | \(=\) | \( - 2^{5} \cdot 7589 \) |
\( J_{10} \) | \(=\) | \(135936\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 59 \) |
\( g_1 \) | \(=\) | \(3538944/59\) | ||
\( g_2 \) | \(=\) | \(153600/59\) | ||
\( g_3 \) | \(=\) | \(-33808/59\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 1 : 1),\, (1 : -1 : 1),\, (-2 : 5 : 1)\)
magma: [C![-2,5,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
magma: [C![-2,0,1],C![-1,0,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
Number of rational Weierstrass points: \(3\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
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);
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{14}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(14\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(14\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2\) | \(0\) | \(2\) |
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - 2z^3\) | \(0\) | \(14\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 15.70753 \) |
Tamagawa product: | \( 28 \) |
Torsion order: | \( 28 \) |
Leading coefficient: | \( 0.560983 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(8\) | \(14\) | \(1 - T\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 3 T + 3 T^{2} )\) | |
\(59\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 59 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.120.3 | yes |
\(7\) | not computed | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
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 \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);