Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = x^5 - 8x^4 + 13x^3 + 83x - 212$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = x^5z - 8x^4z^2 + 13x^3z^3 + 83xz^5 - 212z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 31x^4 + 52x^3 + 2x^2 + 332x - 847$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-212, 83, 0, 13, -8, 1]), R([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-212, 83, 0, 13, -8, 1], R![1, 0, 1]);
sage: X = HyperellipticCurve(R([-847, 332, 2, 52, -31, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(858491\) | \(=\) | \( 409 \cdot 2099 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-858491\) | \(=\) | \( - 409 \cdot 2099 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(17584\) | \(=\) | \( 2^{4} \cdot 7 \cdot 157 \) |
\( I_4 \) | \(=\) | \(547708\) | \(=\) | \( 2^{2} \cdot 7 \cdot 31 \cdot 631 \) |
\( I_6 \) | \(=\) | \(5545501647\) | \(=\) | \( 3 \cdot 7^{2} \cdot 71 \cdot 531331 \) |
\( I_{10} \) | \(=\) | \(-3433964\) | \(=\) | \( - 2^{2} \cdot 409 \cdot 2099 \) |
\( J_2 \) | \(=\) | \(8792\) | \(=\) | \( 2^{3} \cdot 7 \cdot 157 \) |
\( J_4 \) | \(=\) | \(3129518\) | \(=\) | \( 2 \cdot 7 \cdot 23 \cdot 9719 \) |
\( J_6 \) | \(=\) | \(1179953761\) | \(=\) | \( 7^{2} \cdot 197 \cdot 251 \cdot 487 \) |
\( J_8 \) | \(=\) | \(145067638597\) | \(=\) | \( 7^{2} \cdot 2960564053 \) |
\( J_{10} \) | \(=\) | \(-858491\) | \(=\) | \( - 409 \cdot 2099 \) |
\( g_1 \) | \(=\) | \(-52533749281767391232/858491\) | ||
\( g_2 \) | \(=\) | \(-2126867779553219584/858491\) | ||
\( g_3 \) | \(=\) | \(-91209557279331904/858491\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points: \((1 : 0 : 0),\, (4 : -8 : 1),\, (4 : -9 : 1)\)
magma: [C![1,0,0],C![4,-9,1],C![4,-8,1]]; // minimal model
magma: [C![1,0,0],C![4,-1,1],C![4,1,1]]; // simplified model
Number of rational Weierstrass points: \(1\)
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: \(\Z/{5}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((4 : -8 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-8z^3\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((4 : -8 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-8z^3\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((4 : 1 : 1) - (1 : 0 : 0)\) | \(x - 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 15z^3\) | \(0\) | \(5\) |
2-torsion field: 5.3.3433964.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 5.049862 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 1.817950 \) |
Analytic order of Ш: | \( 9 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(409\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 25 T + 409 T^{2} )\) | |
\(2099\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 60 T + 2099 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.6.1 | no |
\(5\) | not computed | yes |
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);