Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x + 1)y = x^5 - 9x^3 + 7x^2 + 13x - 14$ | (homogenize, simplify) |
| $y^2 + (xz^2 + z^3)y = x^5z - 9x^3z^3 + 7x^2z^4 + 13xz^5 - 14z^6$ | (dehomogenize, simplify) |
| $y^2 = 4x^5 - 36x^3 + 29x^2 + 54x - 55$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, 13, 7, -9, 0, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-14, 13, 7, -9, 0, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([-55, 54, 29, -36, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(610423\) | \(=\) | \( 11 \cdot 211 \cdot 263 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-610423\) | \(=\) | \( - 11 \cdot 211 \cdot 263 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(4104\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 19 \) |
| \( I_4 \) | \(=\) | \(8904\) | \(=\) | \( 2^{3} \cdot 3 \cdot 7 \cdot 53 \) |
| \( I_6 \) | \(=\) | \(12533931\) | \(=\) | \( 3^{2} \cdot 769 \cdot 1811 \) |
| \( I_{10} \) | \(=\) | \(-2441692\) | \(=\) | \( - 2^{2} \cdot 11 \cdot 211 \cdot 263 \) |
| \( J_2 \) | \(=\) | \(2052\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 19 \) |
| \( J_4 \) | \(=\) | \(173962\) | \(=\) | \( 2 \cdot 86981 \) |
| \( J_6 \) | \(=\) | \(19454065\) | \(=\) | \( 5 \cdot 89 \cdot 43717 \) |
| \( J_8 \) | \(=\) | \(2414240984\) | \(=\) | \( 2^{3} \cdot 301780123 \) |
| \( J_{10} \) | \(=\) | \(-610423\) | \(=\) | \( - 11 \cdot 211 \cdot 263 \) |
| \( g_1 \) | \(=\) | \(-36382017816364032/610423\) | ||
| \( g_2 \) | \(=\) | \(-1503095107936896/610423\) | ||
| \( g_3 \) | \(=\) | \(-81915309311760/610423\) |
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
Known points: \((1 : 0 : 0),\, (2 : 0 : 1),\, (2 : -3 : 1)\)
magma: [C![1,0,0],C![2,-3,1],C![2,0,1]]; // minimal model
magma: [C![1,0,0],C![2,-3,1],C![2,3,1]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + 2z^3\) | \(0.687204\) | \(\infty\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.648883\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + 2z^3\) | \(0.687204\) | \(\infty\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.648883\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 + 5z^3\) | \(0.687204\) | \(\infty\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + z^3\) | \(0.648883\) | \(\infty\) |
2-torsion field: 5.3.2441692.1
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.397075 \) |
| Real period: | \( 11.34796 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 4.505994 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 11 T^{2} )\) |  |
| \(211\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 211 T^{2} )\) | |
| \(263\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 18 T + 263 T^{2} )\) | |
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 |
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);