Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 - x^4 - 3x^3 + 6x^2 - 6x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z - x^4z^2 - 3x^3z^3 + 6x^2z^4 - 6xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 - 4x^4 - 10x^3 + 24x^2 - 24x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -6, 6, -3, -1, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -6, 6, -3, -1, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, -24, 24, -10, -4, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2457\) | \(=\) | \( 3^{3} \cdot 7 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-95823\) | \(=\) | \( - 3^{4} \cdot 7 \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1932\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 23 \) |
\( I_4 \) | \(=\) | \(57897\) | \(=\) | \( 3^{2} \cdot 7 \cdot 919 \) |
\( I_6 \) | \(=\) | \(45198315\) | \(=\) | \( 3^{2} \cdot 5 \cdot 73 \cdot 13759 \) |
\( I_{10} \) | \(=\) | \(12265344\) | \(=\) | \( 2^{7} \cdot 3^{4} \cdot 7 \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(483\) | \(=\) | \( 3 \cdot 7 \cdot 23 \) |
\( J_4 \) | \(=\) | \(7308\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 29 \) |
\( J_6 \) | \(=\) | \(-43264\) | \(=\) | \( - 2^{8} \cdot 13^{2} \) |
\( J_8 \) | \(=\) | \(-18575844\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 7 \cdot 19 \cdot 103 \cdot 113 \) |
\( J_{10} \) | \(=\) | \(95823\) | \(=\) | \( 3^{4} \cdot 7 \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(46360978629/169\) | ||
\( g_2 \) | \(=\) | \(1452301788/169\) | ||
\( g_3 \) | \(=\) | \(-947968/9\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((0 : -2 : 1)\) | \((-3 : 13 : 1)\) |
\((4 : -13 : 3)\) | \((4 : -78 : 3)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((0 : -2 : 1)\) | \((-3 : 13 : 1)\) |
\((4 : -13 : 3)\) | \((4 : -78 : 3)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : 0 : 1)\) | \((-3 : 0 : 1)\) | \((0 : -3 : 1)\) | \((0 : 3 : 1)\) |
\((4 : -65 : 3)\) | \((4 : 65 : 3)\) |
magma: [C![-3,13,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-78,3],C![4,-13,3]]; // minimal model
magma: [C![-3,0,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![4,-65,3],C![4,65,3]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.066494\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 4z^3\) | \(0\) | \(2\) |
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 3z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-7xz^2 + 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.066494\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 4z^3\) | \(0\) | \(2\) |
\((-3 : 13 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + 3z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-7xz^2 + 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -3 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.066494\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 12xz^2 + 9z^3\) | \(0\) | \(2\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + 3z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 14xz^2 + 11z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.066494 \) |
Real period: | \( 18.98310 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.315566 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(3\) | \(4\) | \(2\) | \(1 + T\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) | |
\(13\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 6 T + 13 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | 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);