Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^2 + x)y = x^5 + 17x^4 + 76x^3 + 14x^2 - 32x + 3$ | (homogenize, simplify) |
| $y^2 + (x^2z + xz^2)y = x^5z + 17x^4z^2 + 76x^3z^3 + 14x^2z^4 - 32xz^5 + 3z^6$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 + 69x^4 + 306x^3 + 57x^2 - 128x + 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, -32, 14, 76, 17, 1]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, -32, 14, 76, 17, 1], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([12, -128, 57, 306, 69, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(1270\) | \(=\) | \( 2 \cdot 5 \cdot 127 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(325120\) | \(=\) | \( 2^{9} \cdot 5 \cdot 127 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(239204\) | \(=\) | \( 2^{2} \cdot 7 \cdot 8543 \) |
| \( I_4 \) | \(=\) | \(126763297\) | \(=\) | \( 126763297 \) |
| \( I_6 \) | \(=\) | \(10436094933809\) | \(=\) | \( 19 \cdot 87961 \cdot 6244451 \) |
| \( I_{10} \) | \(=\) | \(41615360\) | \(=\) | \( 2^{16} \cdot 5 \cdot 127 \) |
| \( J_2 \) | \(=\) | \(59801\) | \(=\) | \( 7 \cdot 8543 \) |
| \( J_4 \) | \(=\) | \(143724846\) | \(=\) | \( 2 \cdot 3 \cdot 23954141 \) |
| \( J_6 \) | \(=\) | \(437833820176\) | \(=\) | \( 2^{4} \cdot 1009 \cdot 27120529 \) |
| \( J_8 \) | \(=\) | \(1381517230655315\) | \(=\) | \( 5 \cdot 13 \cdot 19 \cdot 1373^{2} \cdot 593401 \) |
| \( J_{10} \) | \(=\) | \(325120\) | \(=\) | \( 2^{9} \cdot 5 \cdot 127 \) |
| \( g_1 \) | \(=\) | \(764790054928595680699001/325120\) | ||
| \( g_2 \) | \(=\) | \(15368348330455841308623/162560\) | ||
| \( g_3 \) | \(=\) | \(97860226229056869361/20320\) |
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 | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 8xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(7xz^2 - 4z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 8xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(7xz^2 - 4z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 8xz - 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z + 15xz^2 - 8z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.129032000.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 1.894603 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 0.473650 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(1\) | \(9\) | \(1\) | \(( 1 + T )( 1 - T + 2 T^{2} )\) | |
| \(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T + 5 T^{2} )\) |  |
| \(127\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 20 T + 127 T^{2} )\) | |
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 |
| \(3\) | 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);