Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = x^5 + 5x^4 - 8x^3 - 48x^2 + x$ | (homogenize, simplify) |
$y^2 + xz^2y = x^5z + 5x^4z^2 - 8x^3z^3 - 48x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 20x^4 - 32x^3 - 191x^2 + 4x$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -48, -8, 5, 1]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -48, -8, 5, 1], R![0, 1]);
sage: X = HyperellipticCurve(R([0, 4, -191, -32, 20, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(111989\) | \(=\) | \( 53 \cdot 2113 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(111989\) | \(=\) | \( 53 \cdot 2113 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(16976\) | \(=\) | \( 2^{4} \cdot 1061 \) |
\( I_4 \) | \(=\) | \(7208980\) | \(=\) | \( 2^{2} \cdot 5 \cdot 19 \cdot 61 \cdot 311 \) |
\( I_6 \) | \(=\) | \(34342392335\) | \(=\) | \( 5 \cdot 16603 \cdot 413689 \) |
\( I_{10} \) | \(=\) | \(447956\) | \(=\) | \( 2^{2} \cdot 53 \cdot 2113 \) |
\( J_2 \) | \(=\) | \(8488\) | \(=\) | \( 2^{3} \cdot 1061 \) |
\( J_4 \) | \(=\) | \(1800426\) | \(=\) | \( 2 \cdot 3 \cdot 101 \cdot 2971 \) |
\( J_6 \) | \(=\) | \(432614081\) | \(=\) | \( 432614081 \) |
\( J_8 \) | \(=\) | \(107623634513\) | \(=\) | \( 107623634513 \) |
\( J_{10} \) | \(=\) | \(111989\) | \(=\) | \( 53 \cdot 2113 \) |
\( g_1 \) | \(=\) | \(44058210592401031168/111989\) | ||
\( g_2 \) | \(=\) | \(1101010317277135872/111989\) | ||
\( g_3 \) | \(=\) | \(31168176376153664/111989\) |
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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)\)
magma: [C![-4,2,1],C![0,0,1],C![1,0,0]]; // minimal model
magma: [C![-4,0,1],C![0,0,1],C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(3\)
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: \(\Z/{2}\Z \oplus \Z/{2}\Z\) (conditional)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-4 : 2 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((-4 : 2 : 1) - (1 : 0 : 0)\) | \(x + 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-4 : 2 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((-4 : 2 : 1) - (1 : 0 : 0)\) | \(x + 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x (x + 4z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((-4 : 0 : 1) - (1 : 0 : 0)\) | \(x + 4z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + 4z^3\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) (lower bound) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1.0 \) |
Real period: | \( 2.154971 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 2.154971 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(53\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 6 T + 53 T^{2} )\) | |
\(2113\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 14 T + 2113 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.120.3 | 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);