Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -x^5 + 3x^4 - 3x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -x^5z + 3x^4z^2 - 3x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^5 + 14x^4 + 2x^3 - 11x^2 - 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -3, 0, 3, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -3, 0, 3, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, -2, -11, 2, 14, -4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(122084\) | \(=\) | \( 2^{2} \cdot 23 \cdot 1327 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-244168\) | \(=\) | \( - 2^{3} \cdot 23 \cdot 1327 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1284\) | \(=\) | \( 2^{2} \cdot 3 \cdot 107 \) |
\( I_4 \) | \(=\) | \(56817\) | \(=\) | \( 3^{2} \cdot 59 \cdot 107 \) |
\( I_6 \) | \(=\) | \(17284905\) | \(=\) | \( 3^{2} \cdot 5 \cdot 11 \cdot 34919 \) |
\( I_{10} \) | \(=\) | \(-31253504\) | \(=\) | \( - 2^{10} \cdot 23 \cdot 1327 \) |
\( J_2 \) | \(=\) | \(321\) | \(=\) | \( 3 \cdot 107 \) |
\( J_4 \) | \(=\) | \(1926\) | \(=\) | \( 2 \cdot 3^{2} \cdot 107 \) |
\( J_6 \) | \(=\) | \(47588\) | \(=\) | \( 2^{2} \cdot 11897 \) |
\( J_8 \) | \(=\) | \(2891568\) | \(=\) | \( 2^{4} \cdot 3 \cdot 107 \cdot 563 \) |
\( J_{10} \) | \(=\) | \(-244168\) | \(=\) | \( - 2^{3} \cdot 23 \cdot 1327 \) |
\( g_1 \) | \(=\) | \(-3408200705601/244168\) | ||
\( g_2 \) | \(=\) | \(-31852343043/122084\) | ||
\( g_3 \) | \(=\) | \(-1225878777/61042\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -1 : 2)\) | \((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((-1 : -2 : 2)\) | \((3 : -1 : 1)\) | \((-2 : -5 : 1)\) |
\((-2 : 6 : 3)\) | \((-2 : -7 : 3)\) | \((-2 : 14 : 1)\) | \((3 : -30 : 1)\) | \((15 : -190 : 1)\) | \((15 : -3201 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
\((-1 : -1 : 2)\) | \((-1 : 2 : 1)\) | \((1 : -2 : 1)\) | \((-1 : -2 : 2)\) | \((3 : -1 : 1)\) | \((-2 : -5 : 1)\) |
\((-2 : 6 : 3)\) | \((-2 : -7 : 3)\) | \((-2 : 14 : 1)\) | \((3 : -30 : 1)\) | \((15 : -190 : 1)\) | \((15 : -3201 : 1)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : -1 : 2)\) | \((-1 : 1 : 2)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((-2 : -13 : 3)\) | \((-2 : 13 : 3)\) |
\((-2 : -19 : 1)\) | \((-2 : 19 : 1)\) | \((3 : -29 : 1)\) | \((3 : 29 : 1)\) | \((15 : -3011 : 1)\) | \((15 : 3011 : 1)\) |
magma: [C![-2,-7,3],C![-2,-5,1],C![-2,6,3],C![-2,14,1],C![-1,-2,2],C![-1,-1,1],C![-1,-1,2],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![3,-30,1],C![3,-1,1],C![15,-3201,1],C![15,-190,1]]; // minimal model
magma: [C![-2,-13,3],C![-2,-19,1],C![-2,13,3],C![-2,19,1],C![-1,-1,2],C![-1,-3,1],C![-1,1,2],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![3,-29,1],C![3,29,1],C![15,-3011,1],C![15,3011,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.359261\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.370218\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.229811\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.359261\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.370218\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.229811\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.359261\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.370218\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.229811\) | \(\infty\) |
2-torsion field: 6.4.15626752.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.023629 \) |
Real period: | \( 18.10638 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.283543 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 23 T^{2} )\) | |
\(1327\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 33 T + 1327 T^{2} )\) |
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);