Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = x^5 - x^4 - 39x^3 - 76x^2 + 24x - 6$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = x^5z - x^4z^2 - 39x^3z^3 - 76x^2z^4 + 24xz^5 - 6z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 3x^4 - 154x^3 - 301x^2 + 98x - 23$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 24, -76, -39, -1, 1]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 24, -76, -39, -1, 1], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-23, 98, -301, -154, -3, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2735\) | \(=\) | \( 5 \cdot 547 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(13675\) | \(=\) | \( 5^{2} \cdot 547 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(71764\) | \(=\) | \( 2^{2} \cdot 7 \cdot 11 \cdot 233 \) |
\( I_4 \) | \(=\) | \(259832017\) | \(=\) | \( 157 \cdot 1654981 \) |
\( I_6 \) | \(=\) | \(4804066188809\) | \(=\) | \( 191 \cdot 61703 \cdot 407633 \) |
\( I_{10} \) | \(=\) | \(1750400\) | \(=\) | \( 2^{7} \cdot 5^{2} \cdot 547 \) |
\( J_2 \) | \(=\) | \(17941\) | \(=\) | \( 7 \cdot 11 \cdot 233 \) |
\( J_4 \) | \(=\) | \(2585311\) | \(=\) | \( 19 \cdot 136069 \) |
\( J_6 \) | \(=\) | \(598781761\) | \(=\) | \( 598781761 \) |
\( J_8 \) | \(=\) | \(1014727651845\) | \(=\) | \( 3 \cdot 5 \cdot 67648510123 \) |
\( J_{10} \) | \(=\) | \(13675\) | \(=\) | \( 5^{2} \cdot 547 \) |
\( g_1 \) | \(=\) | \(1858802427581887565701/13675\) | ||
\( g_2 \) | \(=\) | \(14929756777053326131/13675\) | ||
\( g_3 \) | \(=\) | \(192735562462946041/13675\) |
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: trivial
magma: MordellWeilGroupGenus2(Jacobian(C));
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 0.310710 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.621420 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(5\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 - 3 T + 5 T^{2} )\) | |
\(547\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 547 T^{2} )\) |
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);