Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = 2x^5 - 27x^4 + 15x^3 - 162x^2 + 27x - 242$ | (homogenize, simplify) |
$y^2 + xz^2y = 2x^5z - 27x^4z^2 + 15x^3z^3 - 162x^2z^4 + 27xz^5 - 242z^6$ | (dehomogenize, simplify) |
$y^2 = 8x^5 - 108x^4 + 60x^3 - 647x^2 + 108x - 968$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-242, 27, -162, 15, -27, 2]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-242, 27, -162, 15, -27, 2], R![0, 1]);
sage: X = HyperellipticCurve(R([-968, 108, -647, 60, -108, 8]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(493528\) | \(=\) | \( 2^{3} \cdot 7^{2} \cdot 1259 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(987056\) | \(=\) | \( 2^{4} \cdot 7^{2} \cdot 1259 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(265464\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 1229 \) |
\( I_4 \) | \(=\) | \(4876857948\) | \(=\) | \( 2^{2} \cdot 3 \cdot 406404829 \) |
\( I_6 \) | \(=\) | \(318019086615348\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 133831 \cdot 66007603 \) |
\( I_{10} \) | \(=\) | \(-3948224\) | \(=\) | \( - 2^{6} \cdot 7^{2} \cdot 1259 \) |
\( J_2 \) | \(=\) | \(132732\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 1229 \) |
\( J_4 \) | \(=\) | \(-78735332\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 337 \cdot 4493 \) |
\( J_6 \) | \(=\) | \(45902102012\) | \(=\) | \( 2^{2} \cdot 13 \cdot 1009 \cdot 874859 \) |
\( J_8 \) | \(=\) | \(-26643675223360\) | \(=\) | \( - 2^{6} \cdot 5 \cdot 13 \cdot 47 \cdot 9439 \cdot 14437 \) |
\( J_{10} \) | \(=\) | \(-987056\) | \(=\) | \( - 2^{4} \cdot 7^{2} \cdot 1259 \) |
\( g_1 \) | \(=\) | \(-2574887205210133239433152/61691\) | ||
\( g_2 \) | \(=\) | \(11507383731692448218736/61691\) | ||
\( g_3 \) | \(=\) | \(-50543331894663215868/61691\) |
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);
|
Rational points
Known 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: \(\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(18x^2 - xz + 54z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(-14xz^2\) | \(0.308357\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(18x^2 - xz + 54z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(-14xz^2\) | \(0.308357\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(18x^2 - xz + 54z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(-27xz^2\) | \(0.308357\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.308357 \) |
Real period: | \( 0.671857 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 3.729099 \) |
Analytic order of Ш: | \( 9 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 - T\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(1259\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 60 T + 1259 T^{2} )\) |
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 |
\(3\) | 3.80.2 | no |
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);