Minimal equation
Minimal equation
Simplified equation
$y^2 + x^2y = x^5 - 22x^4 - 30x^3 + x^2 + 11x + 2$ | (homogenize, simplify) |
$y^2 + x^2zy = x^5z - 22x^4z^2 - 30x^3z^3 + x^2z^4 + 11xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 87x^4 - 120x^3 + 4x^2 + 44x + 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 11, 1, -30, -22, 1]), R([0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 11, 1, -30, -22, 1], R![0, 0, 1]);
sage: X = HyperellipticCurve(R([8, 44, 4, -120, -87, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(258357\) | \(=\) | \( 3 \cdot 11 \cdot 7829 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(258357\) | \(=\) | \( 3 \cdot 11 \cdot 7829 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(24752\) | \(=\) | \( 2^{4} \cdot 7 \cdot 13 \cdot 17 \) |
\( I_4 \) | \(=\) | \(15988612\) | \(=\) | \( 2^{2} \cdot 67 \cdot 59659 \) |
\( I_6 \) | \(=\) | \(110693925423\) | \(=\) | \( 3^{2} \cdot 229 \cdot 53708843 \) |
\( I_{10} \) | \(=\) | \(1033428\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 7829 \) |
\( J_2 \) | \(=\) | \(12376\) | \(=\) | \( 2^{3} \cdot 7 \cdot 13 \cdot 17 \) |
\( J_4 \) | \(=\) | \(3717122\) | \(=\) | \( 2 \cdot 19 \cdot 23 \cdot 4253 \) |
\( J_6 \) | \(=\) | \(1249461841\) | \(=\) | \( 61 \cdot 20482981 \) |
\( J_8 \) | \(=\) | \(411585945333\) | \(=\) | \( 3 \cdot 167 \cdot 821528833 \) |
\( J_{10} \) | \(=\) | \(258357\) | \(=\) | \( 3 \cdot 11 \cdot 7829 \) |
\( g_1 \) | \(=\) | \(290336410647019749376/258357\) | ||
\( g_2 \) | \(=\) | \(7046082395391183872/258357\) | ||
\( g_3 \) | \(=\) | \(191374292674417216/258357\) |
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: \((1 : 0 : 0)\)
magma: [C![1,0,0]]; // minimal model
magma: [C![1,0,0]]; // 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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + xz - 2z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(xz^2 - 2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + xz - 2z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(xz^2 - 2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + xz - 2z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 2xz^2 - 4z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.35243123229072.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 1.766348 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 3.974284 \) |
Analytic order of Ш: | \( 9 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 11 T^{2} )\) | |
\(7829\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 94 T + 7829 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.60.1 | 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);