Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 7x^3 + 17x^2 + 12x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 7x^3z^3 + 17x^2z^4 + 12xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 30x^3 + 68x^2 + 48x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 12, 17, 7]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 12, 17, 7], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, 48, 68, 30, 0, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(148629\) | \(=\) | \( 3 \cdot 13 \cdot 37 \cdot 103 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(445887\) | \(=\) | \( 3^{2} \cdot 13 \cdot 37 \cdot 103 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1620\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( I_4 \) | \(=\) | \(55689\) | \(=\) | \( 3 \cdot 19 \cdot 977 \) |
\( I_6 \) | \(=\) | \(24130629\) | \(=\) | \( 3^{6} \cdot 79 \cdot 419 \) |
\( I_{10} \) | \(=\) | \(57073536\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 13 \cdot 37 \cdot 103 \) |
\( J_2 \) | \(=\) | \(405\) | \(=\) | \( 3^{4} \cdot 5 \) |
\( J_4 \) | \(=\) | \(4514\) | \(=\) | \( 2 \cdot 37 \cdot 61 \) |
\( J_6 \) | \(=\) | \(79668\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 2213 \) |
\( J_8 \) | \(=\) | \(2972336\) | \(=\) | \( 2^{4} \cdot 23 \cdot 41 \cdot 197 \) |
\( J_{10} \) | \(=\) | \(445887\) | \(=\) | \( 3^{2} \cdot 13 \cdot 37 \cdot 103 \) |
\( g_1 \) | \(=\) | \(1210689028125/49543\) | ||
\( g_2 \) | \(=\) | \(900497250/1339\) | ||
\( g_3 \) | \(=\) | \(1451949300/49543\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((0 : -2 : 1)\) | \((-2 : 2 : 1)\) |
\((-2 : 5 : 1)\) | \((-3 : 8 : 2)\) | \((-7 : -8 : 6)\) | \((3 : 10 : 1)\) | \((-4 : 10 : 3)\) | \((-3 : 11 : 2)\) |
\((-4 : 27 : 3)\) | \((3 : -38 : 1)\) | \((-7 : 135 : 6)\) | \((5 : 3868 : 11)\) | \((5 : -5324 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : 1 : 1)\) | \((0 : -2 : 1)\) | \((-2 : 2 : 1)\) |
\((-2 : 5 : 1)\) | \((-3 : 8 : 2)\) | \((-7 : -8 : 6)\) | \((3 : 10 : 1)\) | \((-4 : 10 : 3)\) | \((-3 : 11 : 2)\) |
\((-4 : 27 : 3)\) | \((3 : -38 : 1)\) | \((-7 : 135 : 6)\) | \((5 : 3868 : 11)\) | \((5 : -5324 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -3 : 1)\) | \((0 : 3 : 1)\) | \((-2 : -3 : 1)\) |
\((-2 : 3 : 1)\) | \((-3 : -3 : 2)\) | \((-3 : 3 : 2)\) | \((-4 : -17 : 3)\) | \((-4 : 17 : 3)\) | \((3 : -48 : 1)\) |
\((3 : 48 : 1)\) | \((-7 : -143 : 6)\) | \((-7 : 143 : 6)\) | \((5 : -9192 : 11)\) | \((5 : 9192 : 11)\) |
magma: [C![-7,-8,6],C![-7,135,6],C![-4,10,3],C![-4,27,3],C![-3,8,2],C![-3,11,2],C![-2,2,1],C![-2,5,1],C![-1,0,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![3,-38,1],C![3,10,1],C![5,-5324,11],C![5,3868,11]]; // minimal model
magma: [C![-7,-143,6],C![-7,143,6],C![-4,-17,3],C![-4,17,3],C![-3,-3,2],C![-3,3,2],C![-2,-3,1],C![-2,3,1],C![-1,0,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![3,-48,1],C![3,48,1],C![5,-9192,11],C![5,9192,11]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - 2z^3\) | \(0.508637\) | \(\infty\) |
\((-3 : 11 : 2) - (1 : 0 : 0)\) | \(z (2x + 3z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.261738\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.295814\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - 2z^3\) | \(0.508637\) | \(\infty\) |
\((-3 : 11 : 2) - (1 : 0 : 0)\) | \(z (2x + 3z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 2z^3\) | \(0.261738\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.295814\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (0 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 4xz^2 - 3z^3\) | \(0.508637\) | \(\infty\) |
\((-3 : 3 : 2) - (1 : 1 : 0)\) | \(z (2x + 3z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - 3z^3\) | \(0.261738\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.295814\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.037789 \) |
Real period: | \( 16.98498 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.283705 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 13 T^{2} )\) | |
\(37\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 37 T^{2} )\) | |
\(103\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 13 T + 103 T^{2} )\) |
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);