Minimal equation
Minimal equation
Simplified equation
| $y^2 + x^2y = 15x^5 + 11x^4 - 12x^3 - x^2 + 14x + 6$ | (homogenize, simplify) |
| $y^2 + x^2zy = 15x^5z + 11x^4z^2 - 12x^3z^3 - x^2z^4 + 14xz^5 + 6z^6$ | (dehomogenize, simplify) |
| $y^2 = 60x^5 + 45x^4 - 48x^3 - 4x^2 + 56x + 24$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([6, 14, -1, -12, 11, 15]), R([0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![6, 14, -1, -12, 11, 15], R![0, 0, 1]);
sage: X = HyperellipticCurve(R([24, 56, -4, -48, 45, 60]))
magma: X,pi:= SimplifiedModel(C);
Invariants
| Conductor: | \( N \) | \(=\) | \(32400\) | \(=\) | \( 2^{4} \cdot 3^{4} \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-486000\) | \(=\) | \( - 2^{4} \cdot 3^{5} \cdot 5^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(12592\) | \(=\) | \( 2^{4} \cdot 787 \) |
| \( I_4 \) | \(=\) | \(1900\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 19 \) |
| \( I_6 \) | \(=\) | \(7833900\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 26113 \) |
| \( I_{10} \) | \(=\) | \(-8000\) | \(=\) | \( - 2^{6} \cdot 5^{3} \) |
| \( J_2 \) | \(=\) | \(18888\) | \(=\) | \( 2^{3} \cdot 3 \cdot 787 \) |
| \( J_4 \) | \(=\) | \(14862006\) | \(=\) | \( 2 \cdot 3^{2} \cdot 353 \cdot 2339 \) |
| \( J_6 \) | \(=\) | \(15589640196\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 13 \cdot 43 \cdot 774679 \) |
| \( J_8 \) | \(=\) | \(18394475419503\) | \(=\) | \( 3^{3} \cdot 19 \cdot 107 \cdot 16763 \cdot 19991 \) |
| \( J_{10} \) | \(=\) | \(-486000\) | \(=\) | \( - 2^{4} \cdot 3^{5} \cdot 5^{3} \) |
| \( g_1 \) | \(=\) | \(-618306218132903936/125\) | ||
| \( g_2 \) | \(=\) | \(-25757819662387264/125\) | ||
| \( g_3 \) | \(=\) | \(-4291439937136144/375\) |
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: \((1 : 0 : 0),\, (-3 : -18 : 4)\)
magma: [C![-3,-18,4],C![1,0,0]]; // minimal model
magma: [C![-3,0,4],C![1,0,0]]; // simplified model
Number of rational Weierstrass points: \(2\)
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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + xz - z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(-8xz^2 - 7z^3\) | \(0.161598\) | \(\infty\) |
| \((-3 : -18 : 4) - (1 : 0 : 0)\) | \(4x + 3z\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(-9z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + xz - z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(-8xz^2 - 7z^3\) | \(0.161598\) | \(\infty\) |
| \((-3 : -18 : 4) - (1 : 0 : 0)\) | \(4x + 3z\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(-9z^3\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + xz - z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(x^2z - 16xz^2 - 14z^3\) | \(0.161598\) | \(\infty\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x + 3z\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(x^2z - 18z^3\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.161598 \) |
| Real period: | \( 11.36554 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 1.836656 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(4\) | \(4\) | \(1\) | \(1 - T + 2 T^{2}\) | |
| \(3\) | \(4\) | \(5\) | \(2\) | \(1 + T\) |  |
| \(5\) | \(2\) | \(3\) | \(2\) | \(1 + 5 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.30.3 | 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);