Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + 1)y = 2x^4 + x^3 - 2x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + z^3)y = 2x^4z^2 + x^3z^3 - 2x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 9x^4 + 6x^3 - 6x^2 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 1, 2]), R([1, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 1, 2], R![1, 0, 1, 1]);
sage: X = HyperellipticCurve(R([1, -4, -6, 6, 9, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(39993\) | \(=\) | \( 3 \cdot 13331 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(119979\) | \(=\) | \( 3^{2} \cdot 13331 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(260\) | \(=\) | \( 2^{2} \cdot 5 \cdot 13 \) |
\( I_4 \) | \(=\) | \(19033\) | \(=\) | \( 7 \cdot 2719 \) |
\( I_6 \) | \(=\) | \(674317\) | \(=\) | \( 7 \cdot 96331 \) |
\( I_{10} \) | \(=\) | \(15357312\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 13331 \) |
\( J_2 \) | \(=\) | \(65\) | \(=\) | \( 5 \cdot 13 \) |
\( J_4 \) | \(=\) | \(-617\) | \(=\) | \( -617 \) |
\( J_6 \) | \(=\) | \(5589\) | \(=\) | \( 3^{5} \cdot 23 \) |
\( J_8 \) | \(=\) | \(-4351\) | \(=\) | \( - 19 \cdot 229 \) |
\( J_{10} \) | \(=\) | \(119979\) | \(=\) | \( 3^{2} \cdot 13331 \) |
\( g_1 \) | \(=\) | \(1160290625/119979\) | ||
\( g_2 \) | \(=\) | \(-169443625/119979\) | ||
\( g_3 \) | \(=\) | \(2623725/13331\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((-1 : 0 : 2)\) | \((2 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-2 : -3 : 1)\) | \((-2 : 6 : 1)\) |
\((-1 : -9 : 2)\) | \((2 : -15 : 1)\) | \((-3 : -15 : 2)\) | \((-3 : 16 : 2)\) | \((1 : -56 : 5)\) | \((1 : -63 : 6)\) |
\((1 : -75 : 5)\) | \((1 : -160 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((-1 : 0 : 2)\) | \((2 : 2 : 1)\) | \((1 : -3 : 1)\) | \((-2 : -3 : 1)\) | \((-2 : 6 : 1)\) |
\((-1 : -9 : 2)\) | \((2 : -15 : 1)\) | \((-3 : -15 : 2)\) | \((-3 : 16 : 2)\) | \((1 : -56 : 5)\) | \((1 : -63 : 6)\) |
\((1 : -75 : 5)\) | \((1 : -160 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((-2 : -9 : 1)\) | \((-2 : 9 : 1)\) | \((-1 : -9 : 2)\) | \((-1 : 9 : 2)\) |
\((2 : -17 : 1)\) | \((2 : 17 : 1)\) | \((1 : -19 : 5)\) | \((1 : 19 : 5)\) | \((-3 : -31 : 2)\) | \((-3 : 31 : 2)\) |
\((1 : -97 : 6)\) | \((1 : 97 : 6)\) |
magma: [C![-3,-15,2],C![-3,16,2],C![-2,-3,1],C![-2,6,1],C![-1,-9,2],C![-1,-1,1],C![-1,0,1],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-160,6],C![1,-75,5],C![1,-63,6],C![1,-56,5],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-15,1],C![2,2,1]]; // minimal model
magma: [C![-3,-31,2],C![-3,31,2],C![-2,-9,1],C![-2,9,1],C![-1,-9,2],C![-1,-1,1],C![-1,1,1],C![-1,9,2],C![0,-1,1],C![0,1,1],C![1,-97,6],C![1,-19,5],C![1,97,6],C![1,19,5],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![2,-17,1],C![2,17,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.390167\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.456379\) | \(\infty\) |
\((-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0.094506\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.390167\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.456379\) | \(\infty\) |
\((-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - 2z^3\) | \(0.094506\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 3 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + z^3\) | \(0.390167\) | \(\infty\) |
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + z^3\) | \(0.456379\) | \(\infty\) |
\((-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z - 2xz^2 - 3z^3\) | \(0.094506\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.013478 \) |
Real period: | \( 18.81218 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.507135 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(13331\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 70 T + 13331 T^{2} )\) |
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);