Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^5 + x^4 - 3x^3 - 2x^2 + x + 2$ | (homogenize, simplify) |
$y^2 + z^3y = x^5z + x^4z^2 - 3x^3z^3 - 2x^2z^4 + xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 4x^4 - 12x^3 - 8x^2 + 4x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, -2, -3, 1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, -2, -3, 1, 1], R![1]);
sage: X = HyperellipticCurve(R([9, 4, -8, -12, 4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(228177\) | \(=\) | \( 3^{6} \cdot 313 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-684531\) | \(=\) | \( - 3^{7} \cdot 313 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(168\) | \(=\) | \( 2^{3} \cdot 3 \cdot 7 \) |
\( I_4 \) | \(=\) | \(720\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5 \) |
\( I_6 \) | \(=\) | \(-33528\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 11 \cdot 127 \) |
\( I_{10} \) | \(=\) | \(-11268\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 313 \) |
\( J_2 \) | \(=\) | \(252\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \) |
\( J_4 \) | \(=\) | \(1566\) | \(=\) | \( 2 \cdot 3^{3} \cdot 29 \) |
\( J_6 \) | \(=\) | \(213228\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5923 \) |
\( J_8 \) | \(=\) | \(12820275\) | \(=\) | \( 3^{4} \cdot 5^{2} \cdot 13 \cdot 487 \) |
\( J_{10} \) | \(=\) | \(-684531\) | \(=\) | \( - 3^{7} \cdot 313 \) |
\( g_1 \) | \(=\) | \(-464679936/313\) | ||
\( g_2 \) | \(=\) | \(-11458944/313\) | ||
\( g_3 \) | \(=\) | \(-18574528/939\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : 0 : 1)\) |
\((0 : -2 : 1)\) | \((-2 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((2 : 4 : 1)\) | \((2 : -5 : 1)\) | \((5 : -26 : 4)\) |
\((5 : -38 : 4)\) | \((7 : 134 : 1)\) | \((7 : -135 : 1)\) | \((4 : 690 : 9)\) | \((4 : -1419 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : 0 : 1)\) |
\((0 : -2 : 1)\) | \((-2 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((2 : 4 : 1)\) | \((2 : -5 : 1)\) | \((5 : -26 : 4)\) |
\((5 : -38 : 4)\) | \((7 : 134 : 1)\) | \((7 : -135 : 1)\) | \((4 : 690 : 9)\) | \((4 : -1419 : 9)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-2 : -1 : 1)\) | \((-2 : 1 : 1)\) | \((0 : -3 : 1)\) |
\((0 : 3 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((2 : -9 : 1)\) | \((2 : 9 : 1)\) | \((5 : -12 : 4)\) |
\((5 : 12 : 4)\) | \((7 : -269 : 1)\) | \((7 : 269 : 1)\) | \((4 : -2109 : 9)\) | \((4 : 2109 : 9)\) |
magma: [C![-2,-1,1],C![-2,0,1],C![-1,-2,1],C![-1,1,1],C![0,-2,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,4,1],C![4,-1419,9],C![4,690,9],C![5,-38,4],C![5,-26,4],C![7,-135,1],C![7,134,1]]; // minimal model
magma: [C![-2,-1,1],C![-2,1,1],C![-1,-3,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![2,-9,1],C![2,9,1],C![4,-2109,9],C![4,2109,9],C![5,-12,4],C![5,12,4],C![7,-269,1],C![7,269,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 2z^3\) | \(0.766357\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.395818\) | \(\infty\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.209929\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -2 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 2z^3\) | \(0.766357\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.395818\) | \(\infty\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.209929\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -3 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - 3z^3\) | \(0.766357\) | \(\infty\) |
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.395818\) | \(\infty\) |
\((-1 : 3 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.209929\) | \(\infty\) |
2-torsion field: 5.3.1216944.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.054008 \) |
Real period: | \( 14.64445 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.581840 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(6\) | \(7\) | \(2\) | \(1 + T\) | |
\(313\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 10 T + 313 T^{2} )\) |
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);