Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -2x^4 + 9x^2 + 9x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -2x^4z^2 + 9x^2z^4 + 9xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^4 + 2x^3 + 36x^2 + 36x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 9, 9, 0, -2]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 9, 9, 0, -2], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, 36, 36, 2, -8, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(15957\) | \(=\) | \( 3^{4} \cdot 197 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-143613\) | \(=\) | \( - 3^{6} \cdot 197 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(412\) | \(=\) | \( 2^{2} \cdot 103 \) |
\( I_4 \) | \(=\) | \(4641\) | \(=\) | \( 3 \cdot 7 \cdot 13 \cdot 17 \) |
\( I_6 \) | \(=\) | \(548591\) | \(=\) | \( 548591 \) |
\( I_{10} \) | \(=\) | \(-75648\) | \(=\) | \( - 2^{7} \cdot 3 \cdot 197 \) |
\( J_2 \) | \(=\) | \(309\) | \(=\) | \( 3 \cdot 103 \) |
\( J_4 \) | \(=\) | \(2238\) | \(=\) | \( 2 \cdot 3 \cdot 373 \) |
\( J_6 \) | \(=\) | \(11956\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 61 \) |
\( J_8 \) | \(=\) | \(-328560\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 5 \cdot 37^{2} \) |
\( J_{10} \) | \(=\) | \(-143613\) | \(=\) | \( - 3^{6} \cdot 197 \) |
\( g_1 \) | \(=\) | \(-11592740743/591\) | ||
\( g_2 \) | \(=\) | \(-815174342/1773\) | ||
\( g_3 \) | \(=\) | \(-126841204/15957\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((0 : -2 : 1)\) | \((-2 : 3 : 1)\) |
\((3 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-3 : 8 : 2)\) | \((-3 : 11 : 2)\) | \((3 : -26 : 1)\) | \((-1 : 50 : 5)\) |
\((-1 : -174 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((0 : -2 : 1)\) | \((-2 : 3 : 1)\) |
\((3 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((-3 : 8 : 2)\) | \((-3 : 11 : 2)\) | \((3 : -26 : 1)\) | \((-1 : 50 : 5)\) |
\((-1 : -174 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : 0 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) | \((0 : -3 : 1)\) |
\((0 : 3 : 1)\) | \((-3 : -3 : 2)\) | \((-3 : 3 : 2)\) | \((3 : -24 : 1)\) | \((3 : 24 : 1)\) | \((-1 : -224 : 5)\) |
\((-1 : 224 : 5)\) |
magma: [C![-3,8,2],C![-3,11,2],C![-2,3,1],C![-2,4,1],C![-1,-174,5],C![-1,0,1],C![-1,50,5],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![3,-26,1],C![3,-2,1]]; // minimal model
magma: [C![-3,-3,2],C![-3,3,2],C![-2,-1,1],C![-2,1,1],C![-1,-224,5],C![-1,0,1],C![-1,224,5],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![3,-24,1],C![3,24,1]]; // simplified model
Number of rational Weierstrass points: \(1\)
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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.146215\) | \(\infty\) |
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.065797\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.146215\) | \(\infty\) |
\((0 : -2 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.065797\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.146215\) | \(\infty\) |
\((0 : -3 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.065797\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.009511 \) |
Real period: | \( 19.28012 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.550119 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(4\) | \(6\) | \(3\) | \(1 + 2 T + 3 T^{2}\) | |
\(197\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 26 T + 197 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | no |
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);