Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -x^5 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -x^5z + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^5 + 2x^4 + 2x^3 + x^2 + 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, 0, 0, -1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, 0, 0, -1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 6, 1, 2, 2, -4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(39017\) | \(=\) | \( 11 \cdot 3547 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-39017\) | \(=\) | \( - 11 \cdot 3547 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(604\) | \(=\) | \( 2^{2} \cdot 151 \) |
\( I_4 \) | \(=\) | \(-1943\) | \(=\) | \( - 29 \cdot 67 \) |
\( I_6 \) | \(=\) | \(386715\) | \(=\) | \( 3 \cdot 5 \cdot 7 \cdot 29 \cdot 127 \) |
\( I_{10} \) | \(=\) | \(4994176\) | \(=\) | \( 2^{7} \cdot 11 \cdot 3547 \) |
\( J_2 \) | \(=\) | \(151\) | \(=\) | \( 151 \) |
\( J_4 \) | \(=\) | \(1031\) | \(=\) | \( 1031 \) |
\( J_6 \) | \(=\) | \(-797\) | \(=\) | \( -797 \) |
\( J_8 \) | \(=\) | \(-295827\) | \(=\) | \( - 3 \cdot 7 \cdot 14087 \) |
\( J_{10} \) | \(=\) | \(39017\) | \(=\) | \( 11 \cdot 3547 \) |
\( g_1 \) | \(=\) | \(78502725751/39017\) | ||
\( g_2 \) | \(=\) | \(3549682481/39017\) | ||
\( g_3 \) | \(=\) | \(-18172397/39017\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : 1 : 1)\) | \((1 : 2 : 2)\) | \((1 : -3 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) | \((1 : -15 : 2)\) |
\((3 : -15 : 1)\) | \((3 : -16 : 1)\) | \((-1 : -74 : 6)\) | \((-1 : -105 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : 1 : 1)\) | \((1 : 2 : 2)\) | \((1 : -3 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) | \((1 : -15 : 2)\) |
\((3 : -15 : 1)\) | \((3 : -16 : 1)\) | \((-1 : -74 : 6)\) | \((-1 : -105 : 6)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((3 : -1 : 1)\) | \((3 : 1 : 1)\) |
\((1 : -17 : 2)\) | \((1 : 17 : 2)\) | \((-1 : -31 : 6)\) | \((-1 : 31 : 6)\) |
magma: [C![-1,-105,6],C![-1,-74,6],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-15,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,2,2],C![2,-6,1],C![2,-5,1],C![3,-16,1],C![3,-15,1]]; // minimal model
magma: [C![-1,-31,6],C![-1,31,6],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-17,2],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![1,17,2],C![2,-1,1],C![2,1,1],C![3,-1,1],C![3,1,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.663005\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.286528\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^2z\) | \(0.159398\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.663005\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.286528\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^2z\) | \(0.159398\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (2 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2 - z^3\) | \(0.663005\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.286528\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2x^2z + xz^2 + z^3\) | \(0.159398\) | \(\infty\) |
2-torsion field: 6.4.2497088.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.027876 \) |
Real period: | \( 18.01048 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.502076 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 11 T^{2} )\) | |
\(3547\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 13 T + 3547 T^{2} )\) |
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);