Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - x^5 + 3x^4 - 3x^3 + 3x^2 - x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - x^5z + 3x^4z^2 - 3x^3z^3 + 3x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^5 + 12x^4 - 12x^3 + 12x^2 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 3, -3, 3, -1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 3, -3, 3, -1, 1], R![1]);
sage: X = HyperellipticCurve(R([1, -4, 12, -12, 12, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(88149\) | \(=\) | \( 3 \cdot 29383 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-264447\) | \(=\) | \( - 3^{2} \cdot 29383 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(440\) | \(=\) | \( 2^{3} \cdot 5 \cdot 11 \) |
\( I_4 \) | \(=\) | \(3796\) | \(=\) | \( 2^{2} \cdot 13 \cdot 73 \) |
\( I_6 \) | \(=\) | \(427888\) | \(=\) | \( 2^{4} \cdot 47 \cdot 569 \) |
\( I_{10} \) | \(=\) | \(1057788\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 29383 \) |
\( J_2 \) | \(=\) | \(220\) | \(=\) | \( 2^{2} \cdot 5 \cdot 11 \) |
\( J_4 \) | \(=\) | \(1384\) | \(=\) | \( 2^{3} \cdot 173 \) |
\( J_6 \) | \(=\) | \(15768\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 73 \) |
\( J_8 \) | \(=\) | \(388376\) | \(=\) | \( 2^{3} \cdot 43 \cdot 1129 \) |
\( J_{10} \) | \(=\) | \(264447\) | \(=\) | \( 3^{2} \cdot 29383 \) |
\( g_1 \) | \(=\) | \(515363200000/264447\) | ||
\( g_2 \) | \(=\) | \(14736832000/264447\) | ||
\( g_3 \) | \(=\) | \(84796800/29383\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 1 : 1)\) | \((1 : -2 : 1)\) |
\((-1 : 3 : 1)\) | \((-1 : -4 : 1)\) | \((2 : 7 : 3)\) | \((-2 : 13 : 1)\) | \((-2 : -14 : 1)\) | \((1 : -14 : 5)\) |
\((2 : -34 : 3)\) | \((1 : -111 : 5)\) | \((4 : -209 : 13)\) | \((4 : -1988 : 13)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 1 : 1)\) | \((1 : -2 : 1)\) |
\((-1 : 3 : 1)\) | \((-1 : -4 : 1)\) | \((2 : 7 : 3)\) | \((-2 : 13 : 1)\) | \((-2 : -14 : 1)\) | \((1 : -14 : 5)\) |
\((2 : -34 : 3)\) | \((1 : -111 : 5)\) | \((4 : -209 : 13)\) | \((4 : -1988 : 13)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -3 : 1)\) | \((1 : 3 : 1)\) |
\((-1 : -7 : 1)\) | \((-1 : 7 : 1)\) | \((-2 : -27 : 1)\) | \((-2 : 27 : 1)\) | \((2 : -41 : 3)\) | \((2 : 41 : 3)\) |
\((1 : -97 : 5)\) | \((1 : 97 : 5)\) | \((4 : -1779 : 13)\) | \((4 : 1779 : 13)\) |
magma: [C![-2,-14,1],C![-2,13,1],C![-1,-4,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-111,5],C![1,-14,5],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-34,3],C![2,7,3],C![4,-1988,13],C![4,-209,13]]; // minimal model
magma: [C![-2,-27,1],C![-2,27,1],C![-1,-7,1],C![-1,7,1],C![0,-1,1],C![0,1,1],C![1,-97,5],C![1,97,5],C![1,-3,1],C![1,-2,0],C![1,2,0],C![1,3,1],C![2,-41,3],C![2,41,3],C![4,-1779,13],C![4,1779,13]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0.519044\) | \(\infty\) |
\((1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.334564\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.245498\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0.519044\) | \(\infty\) |
\((1 : 1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.334564\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.245498\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) + (1 : 3 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(4xz^2 - z^3\) | \(0.519044\) | \(\infty\) |
\((1 : 3 : 1) - (1 : -2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 + z^3\) | \(0.334564\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -3 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.245498\) | \(\infty\) |
2-torsion field: 6.0.1880512.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.038489 \) |
Real period: | \( 11.18803 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.861249 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(29383\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 174 T + 29383 T^{2} )\) |
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);