Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 5x^2 - 5x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + x^3z^3 + 5x^2z^4 - 5xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 10x^4 + 6x^3 + 21x^2 - 18x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -5, 5, 1, -3]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -5, 5, 1, -3], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, -18, 21, 6, -10, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(62233\) | \(=\) | \( 62233 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-62233\) | \(=\) | \( -62233 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1668\) | \(=\) | \( 2^{2} \cdot 3 \cdot 139 \) |
\( I_4 \) | \(=\) | \(64473\) | \(=\) | \( 3 \cdot 21491 \) |
\( I_6 \) | \(=\) | \(30534477\) | \(=\) | \( 3 \cdot 29 \cdot 350971 \) |
\( I_{10} \) | \(=\) | \(-7965824\) | \(=\) | \( - 2^{7} \cdot 62233 \) |
\( J_2 \) | \(=\) | \(417\) | \(=\) | \( 3 \cdot 139 \) |
\( J_4 \) | \(=\) | \(4559\) | \(=\) | \( 47 \cdot 97 \) |
\( J_6 \) | \(=\) | \(54933\) | \(=\) | \( 3 \cdot 18311 \) |
\( J_8 \) | \(=\) | \(530645\) | \(=\) | \( 5 \cdot 106129 \) |
\( J_{10} \) | \(=\) | \(-62233\) | \(=\) | \( -62233 \) |
\( g_1 \) | \(=\) | \(-12608989261857/62233\) | ||
\( g_2 \) | \(=\) | \(-330580899567/62233\) | ||
\( g_3 \) | \(=\) | \(-9552244437/62233\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-1 : -2 : 1)\) |
\((1 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) | \((-3 : 14 : 1)\) | \((-3 : 15 : 1)\) |
\((-4 : -27 : 3)\) | \((5 : -53 : 2)\) | \((-4 : 100 : 3)\) | \((5 : -100 : 2)\) | \((11 : -891 : 6)\) | \((11 : -1052 : 6)\) |
\((73 : -251641 : 52)\) | \((73 : -475376 : 52)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-1 : -2 : 1)\) |
\((1 : -2 : 1)\) | \((-1 : 3 : 1)\) | \((2 : -5 : 1)\) | \((2 : -6 : 1)\) | \((-3 : 14 : 1)\) | \((-3 : 15 : 1)\) |
\((-4 : -27 : 3)\) | \((5 : -53 : 2)\) | \((-4 : 100 : 3)\) | \((5 : -100 : 2)\) | \((11 : -891 : 6)\) | \((11 : -1052 : 6)\) |
\((73 : -251641 : 52)\) | \((73 : -475376 : 52)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((-3 : -1 : 1)\) | \((-3 : 1 : 1)\) | \((-1 : -5 : 1)\) | \((-1 : 5 : 1)\) |
\((5 : -47 : 2)\) | \((5 : 47 : 2)\) | \((-4 : -127 : 3)\) | \((-4 : 127 : 3)\) | \((11 : -161 : 6)\) | \((11 : 161 : 6)\) |
\((73 : -223735 : 52)\) | \((73 : 223735 : 52)\) |
magma: [C![-4,-27,3],C![-4,100,3],C![-3,14,1],C![-3,15,1],C![-1,-2,1],C![-1,3,1],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![2,-6,1],C![2,-5,1],C![5,-100,2],C![5,-53,2],C![11,-1052,6],C![11,-891,6],C![73,-475376,52],C![73,-251641,52]]; // minimal model
magma: [C![-4,-127,3],C![-4,127,3],C![-3,-1,1],C![-3,1,1],C![-1,-5,1],C![-1,5,1],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![2,-1,1],C![2,1,1],C![5,-47,2],C![5,47,2],C![11,-161,6],C![11,161,6],C![73,-223735,52],C![73,223735,52]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2\) | \(0.562559\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.285327\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.212996\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2\) | \(0.562559\) | \(\infty\) |
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.285327\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.212996\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 5xz^2 + z^3\) | \(0.562559\) | \(\infty\) |
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - 3z^3\) | \(0.285327\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.212996\) | \(\infty\) |
2-torsion field: 6.4.3982912.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.026759 \) |
Real period: | \( 21.72187 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.581268 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(62233\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 146 T + 62233 T^{2} )\) |
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);