Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - x^5 - x^4 - x^3 + x^2 + x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - x^5z - x^4z^2 - x^3z^3 + x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^5 - 4x^4 - 4x^3 + 4x^2 + 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -1, -1, -1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -1, -1, -1, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 4, 4, -4, -4, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(66161\) | \(=\) | \( 66161 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(66161\) | \(=\) | \( 66161 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(312\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \) |
\( I_4 \) | \(=\) | \(5940\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \) |
\( I_6 \) | \(=\) | \(589248\) | \(=\) | \( 2^{6} \cdot 3^{3} \cdot 11 \cdot 31 \) |
\( I_{10} \) | \(=\) | \(-264644\) | \(=\) | \( - 2^{2} \cdot 66161 \) |
\( J_2 \) | \(=\) | \(156\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13 \) |
\( J_4 \) | \(=\) | \(24\) | \(=\) | \( 2^{3} \cdot 3 \) |
\( J_6 \) | \(=\) | \(-13784\) | \(=\) | \( - 2^{3} \cdot 1723 \) |
\( J_8 \) | \(=\) | \(-537720\) | \(=\) | \( - 2^{3} \cdot 3 \cdot 5 \cdot 4481 \) |
\( J_{10} \) | \(=\) | \(-66161\) | \(=\) | \( -66161 \) |
\( g_1 \) | \(=\) | \(-92389579776/66161\) | ||
\( g_2 \) | \(=\) | \(-91113984/66161\) | ||
\( g_3 \) | \(=\) | \(335447424/66161\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : -7 : 3)\) | \((-2 : 9 : 1)\) | \((-2 : -10 : 1)\) | \((2 : 11 : 3)\) |
\((-1 : -20 : 3)\) | \((2 : -38 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : -7 : 3)\) | \((-2 : 9 : 1)\) | \((-2 : -10 : 1)\) | \((2 : 11 : 3)\) |
\((-1 : -20 : 3)\) | \((2 : -38 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((-1 : -13 : 3)\) | \((-1 : 13 : 3)\) | \((-2 : -19 : 1)\) | \((-2 : 19 : 1)\) |
\((2 : -49 : 3)\) | \((2 : 49 : 3)\) |
magma: [C![-2,-10,1],C![-2,9,1],C![-1,-20,3],C![-1,-7,3],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-38,3],C![2,11,3]]; // minimal model
magma: [C![-2,-19,1],C![-2,19,1],C![-1,-13,3],C![-1,13,3],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-49,3],C![2,49,3]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.564970\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.366414\) | \(\infty\) |
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.231439\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.564970\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.366414\) | \(\infty\) |
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.231439\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.564970\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -2 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 - z^3\) | \(0.366414\) | \(\infty\) |
\(2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 + z^3\) | \(0.231439\) | \(\infty\) |
2-torsion field: 6.2.4234304.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.042579 \) |
Real period: | \( 18.10343 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.770840 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(66161\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 478 T + 66161 T^{2} )\) |
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);