Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2)y = -4x^4 + 9x^3 - 9x + 4$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z)y = -4x^4z^2 + 9x^3z^3 - 9xz^5 + 4z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 - 15x^4 + 36x^3 - 36x + 16$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, -9, 0, 9, -4]), R([0, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -9, 0, 9, -4], R![0, 0, 1, 1]);
sage: X = HyperellipticCurve(R([16, -36, 0, 36, -15, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(15334\) | \(=\) | \( 2 \cdot 11 \cdot 17 \cdot 41 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-521356\) | \(=\) | \( - 2^{2} \cdot 11 \cdot 17^{2} \cdot 41 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
\( I_4 \) | \(=\) | \(135432\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 11 \cdot 19 \) |
\( I_6 \) | \(=\) | \(-63999477\) | \(=\) | \( - 3^{4} \cdot 47 \cdot 16811 \) |
\( I_{10} \) | \(=\) | \(-2085424\) | \(=\) | \( - 2^{4} \cdot 11 \cdot 17^{2} \cdot 41 \) |
\( J_2 \) | \(=\) | \(132\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \) |
\( J_4 \) | \(=\) | \(-21846\) | \(=\) | \( - 2 \cdot 3 \cdot 11 \cdot 331 \) |
\( J_6 \) | \(=\) | \(7944017\) | \(=\) | \( 7944017 \) |
\( J_8 \) | \(=\) | \(142840632\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \cdot 19 \cdot 28477 \) |
\( J_{10} \) | \(=\) | \(-521356\) | \(=\) | \( - 2^{2} \cdot 11 \cdot 17^{2} \cdot 41 \) |
\( g_1 \) | \(=\) | \(-910787328/11849\) | ||
\( g_2 \) | \(=\) | \(1141934112/11849\) | ||
\( g_3 \) | \(=\) | \(-3145830732/11849\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((1 : -2 : 1)\) | \((7 : -156 : 11)\) | \((7 : -408 : 10)\) | \((7 : -425 : 10)\) | \((7 : -726 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((1 : -2 : 1)\) | \((7 : -156 : 11)\) | \((7 : -408 : 10)\) | \((7 : -425 : 10)\) | \((7 : -726 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : 0 : 1)\) | \((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((0 : -4 : 1)\) |
\((0 : 4 : 1)\) | \((7 : -17 : 10)\) | \((7 : 17 : 10)\) | \((7 : -570 : 11)\) | \((7 : 570 : 11)\) |
magma: [C![-1,0,1],C![0,-2,1],C![0,2,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![7,-726,11],C![7,-425,10],C![7,-408,10],C![7,-156,11]]; // minimal model
magma: [C![-1,0,1],C![0,-4,1],C![0,4,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![7,-570,11],C![7,-17,10],C![7,17,10],C![7,570,11]]; // 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: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.406428\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.115629\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 5xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-12xz^2 + 8z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.406428\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.115629\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 5xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-12xz^2 + 8z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z\) | \(0.406428\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z\) | \(0.115629\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 5xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 24xz^2 + 16z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.133431056.2
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.046725 \) |
Real period: | \( 13.31796 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.622286 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 11 T^{2} )\) | |
\(17\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + 6 T + 17 T^{2} )\) | |
\(41\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 41 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.60.1 | 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);