Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2)y = x^4 + x^3 - x - 2$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z)y = x^4z^2 + x^3z^3 - xz^5 - 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 5x^4 + 4x^3 - 4x - 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -1, 0, 1, 1]), R([0, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -1, 0, 1, 1], R![0, 0, 1, 1]);
sage: X = HyperellipticCurve(R([-8, -4, 0, 4, 5, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(847\) | \(=\) | \( 7 \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(9317\) | \(=\) | \( 7 \cdot 11^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(424\) | \(=\) | \( 2^{3} \cdot 53 \) |
\( I_4 \) | \(=\) | \(3520\) | \(=\) | \( 2^{6} \cdot 5 \cdot 11 \) |
\( I_6 \) | \(=\) | \(581427\) | \(=\) | \( 3^{2} \cdot 7 \cdot 11 \cdot 839 \) |
\( I_{10} \) | \(=\) | \(37268\) | \(=\) | \( 2^{2} \cdot 7 \cdot 11^{3} \) |
\( J_2 \) | \(=\) | \(212\) | \(=\) | \( 2^{2} \cdot 53 \) |
\( J_4 \) | \(=\) | \(1286\) | \(=\) | \( 2 \cdot 643 \) |
\( J_6 \) | \(=\) | \(-7999\) | \(=\) | \( - 19 \cdot 421 \) |
\( J_8 \) | \(=\) | \(-837396\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 7 \cdot 3323 \) |
\( J_{10} \) | \(=\) | \(9317\) | \(=\) | \( 7 \cdot 11^{3} \) |
\( g_1 \) | \(=\) | \(428232184832/9317\) | ||
\( g_2 \) | \(=\) | \(12253172608/9317\) | ||
\( g_3 \) | \(=\) | \(-359507056/9317\) |
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
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)\)
magma: [C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
magma: [C![1,-1,0],C![1,0,1],C![1,1,0]]; // 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: \(\Z/{8}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(5xz^2 + 2z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(5xz^2 + 2z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 + 3xz + 6z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 + x^2z + 10xz^2 + 4z^3\) | \(0\) | \(8\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 9.983399 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.311981 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 7 T^{2} )\) | |
\(11\) | \(2\) | \(3\) | \(2\) | \(1 + 4 T + 11 T^{2}\) |
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);