Minimal equation
Minimal equation
Simplified equation
| $y^2 + x^3y = x^5 - 4x^4 - 6x^3 + 33x^2 - 36x + 12$ | (homogenize, simplify) |
| $y^2 + x^3y = x^5z - 4x^4z^2 - 6x^3z^3 + 33x^2z^4 - 36xz^5 + 12z^6$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 4x^5 - 16x^4 - 24x^3 + 132x^2 - 144x + 48$ | (minimize, homogenize) |
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![12, -36, 33, -6, -4, 1], R![0, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([12, -36, 33, -6, -4, 1]), R([0, 0, 0, 1]));
magma: X,pi:= SimplifiedModel(C);
sage: X = HyperellipticCurve(R([48, -144, 132, -24, -16, 4, 1]))
Invariants
| Conductor: | \( N \) | = | \(864\) | = | \( 2^{5} \cdot 3^{3} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(864,2),R![1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | = | \(-221184\) | = | \( - 2^{13} \cdot 3^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | = | \(2688\) | = | \( 2^{7} \cdot 3 \cdot 7 \) |
| \( I_4 \) | = | \(8847360\) | = | \( 2^{16} \cdot 3^{3} \cdot 5 \) |
| \( I_6 \) | = | \(-866009088\) | = | \( - 2^{14} \cdot 3^{2} \cdot 7 \cdot 839 \) |
| \( I_{10} \) | = | \(-905969664\) | = | \( - 2^{25} \cdot 3^{3} \) |
| \( J_2 \) | = | \(336\) | = | \( 2^{4} \cdot 3 \cdot 7 \) |
| \( J_4 \) | = | \(-87456\) | = | \( - 2^{5} \cdot 3 \cdot 911 \) |
| \( J_6 \) | = | \(10192896\) | = | \( 2^{11} \cdot 3^{2} \cdot 7 \cdot 79 \) |
| \( J_8 \) | = | \(-1055934720\) | = | \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 71 \cdot 1291 \) |
| \( J_{10} \) | = | \(-221184\) | = | \( - 2^{13} \cdot 3^{3} \) |
| \( g_1 \) | = | \(-19361664\) | ||
| \( g_2 \) | = | \(14998704\) | ||
| \( g_3 \) | = | \(-5202624\) |
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)\)
magma: [C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]];
Number of rational Weierstrass points: \(0\)
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: \(\Z/{12}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(2 \cdot(1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z)^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0\) | \(12\) |
2-torsion field: 8.0.47775744.4
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(1\) |
| Regulator: | \( 1 \) |
| Real period: | \( 18.14296 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 12 \) |
| Leading coefficient: | \( 0.377978 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor |
|---|---|---|---|---|
| \(2\) | \(13\) | \(5\) | \(3\) | \(1\) |
| \(3\) | \(3\) | \(3\) | \(1\) | \(1 + T\) |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $N(G_{1,3})$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 36.a2
Elliptic curve 24.a2
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(8\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-3}) \) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |