Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = x^6 - x^4 - x^3 + x$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = x^6 - x^4z^2 - x^3z^3 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 3x^4 - 4x^3 + 2x^2 + 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -1, -1, 0, 1]), R([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -1, -1, 0, 1], R![1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 4, 2, -4, -3, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(8588\) | \(=\) | \( 2^{2} \cdot 19 \cdot 113 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(34352\) | \(=\) | \( 2^{4} \cdot 19 \cdot 113 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(192\) | \(=\) | \( 2^{6} \cdot 3 \) |
\( I_4 \) | \(=\) | \(1644\) | \(=\) | \( 2^{2} \cdot 3 \cdot 137 \) |
\( I_6 \) | \(=\) | \(150180\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 2503 \) |
\( I_{10} \) | \(=\) | \(-137408\) | \(=\) | \( - 2^{6} \cdot 19 \cdot 113 \) |
\( J_2 \) | \(=\) | \(96\) | \(=\) | \( 2^{5} \cdot 3 \) |
\( J_4 \) | \(=\) | \(110\) | \(=\) | \( 2 \cdot 5 \cdot 11 \) |
\( J_6 \) | \(=\) | \(-7332\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 13 \cdot 47 \) |
\( J_8 \) | \(=\) | \(-178993\) | \(=\) | \( - 17 \cdot 10529 \) |
\( J_{10} \) | \(=\) | \(-34352\) | \(=\) | \( - 2^{4} \cdot 19 \cdot 113 \) |
\( g_1 \) | \(=\) | \(-509607936/2147\) | ||
\( g_2 \) | \(=\) | \(-6082560/2147\) | ||
\( g_3 \) | \(=\) | \(4223232/2147\) |
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
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -2 : 1)\) | \((1 : -2 : 1)\) | \((-1 : -14 : 3)\) | \((-2 : -14 : 3)\) | \((-1 : -16 : 3)\) | \((-2 : -25 : 3)\) |
\((-40 : 37665 : 23)\) | \((-40 : -86632 : 23)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -2 : 1)\) | \((1 : -2 : 1)\) | \((-1 : -14 : 3)\) | \((-2 : -14 : 3)\) | \((-1 : -16 : 3)\) | \((-2 : -25 : 3)\) |
\((-40 : 37665 : 23)\) | \((-40 : -86632 : 23)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((-1 : -2 : 3)\) | \((-1 : 2 : 3)\) | \((-2 : -11 : 3)\) | \((-2 : 11 : 3)\) |
\((-40 : -124297 : 23)\) | \((-40 : 124297 : 23)\) |
magma: [C![-40,-86632,23],C![-40,37665,23],C![-2,-25,3],C![-2,-14,3],C![-1,-16,3],C![-1,-14,3],C![-1,-2,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // minimal model
magma: [C![-40,-124297,23],C![-40,124297,23],C![-2,-11,3],C![-2,11,3],C![-1,-2,3],C![-1,2,3],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-2,1],C![1,-2,0],C![1,2,1],C![1,2,0]]; // simplified model
Number of rational Weierstrass points: \(0\)
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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.167538\) | \(\infty\) |
\((-1 : -2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(0.047668\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.167538\) | \(\infty\) |
\((-1 : -2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(0.047668\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 2xz^2 + z^3\) | \(0.167538\) | \(\infty\) |
\((-1 : -2 : 1) + (0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 4xz^2 + z^3\) | \(0.047668\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.007661 \) |
Real period: | \( 19.41362 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.446187 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(3\) | \(1 + T + 2 T^{2}\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 19 T^{2} )\) | |
\(113\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 12 T + 113 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);