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 | 1 | initial version |
In what form do you want it? As a q-series? Or do you want j as a rational function of it?
If the former, I don't think there exists an algorithm, though there are methods.
If the latter, Sage does know a few, namely for $X_0(\ell)$ for $\ell=2,3,5,7,13$::
{{{
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import Fricke_modules
sage: Fricke_modules.keys()
[3, 13, 5, 7]
sage: Fricke_modules[3]
(t^4 + 36t^3 + 270t^2 + 756t + 729)/t
sage: Fricke_modules[5]
(t^6 + 30t^5 + 315t^4 + 1300t^3 + 1575t^2 + 750t + 125)/t
sage: Fricke_modules[7]
(t^8 + 28t^7 + 322t^6 + 1904t^5 + 5915t^4 + 8624t^3 + 4018t^2 + 748t + 49)/t
sage: Fricke_modules[13]
(t^14 + 26t^13 + 325t^12 + 2548t^11 + 13832t^10 + 54340t^9 + 157118t^8 + 333580t^7 + 509366t^6 + 534820t^5 + 354536t^4 + 124852t^3 + 15145t^2 + 746t + 13)/t
}}}
| | 2 | No.2 Revision |
In what form do you want it? As a q-series? Or do you want j as a rational function of it?
If the former, I don't think there exists an algorithm, though there are methods.
If the latter, Sage does know a few, namely for $X_0(\ell)$ for $\ell=2,3,5,7,13$::
{{{
$\ell=2,3,5,7,13$::
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import Fricke_modules
sage: Fricke_modules.keys()
[3, 13, 5, 7]
sage: Fricke_modules[3]
36t^3 36*t^3 + 270t^2 270*t^2 + 756t 756*t + 729)/t
sage: Fricke_modules[5]
(t^6 + 30t^5 30*t^5 + 315t^4 315*t^4 + 1300t^3 1300*t^3 + 1575t^2 1575*t^2 + 750t 750*t + 125)/t
sage: Fricke_modules[7]
(t^8 + 28t^7 28*t^7 + 322t^6 322*t^6 + 1904t^5 1904*t^5 + 5915t^4 5915*t^4 + 8624t^3 8624*t^3 + 4018t^2 4018*t^2 + 748t 748*t + 49)/t
sage: Fricke_modules[13]
(t^14 + 26t^13 26*t^13 + 325t^12 325*t^12 + 2548t^11 2548*t^11 + 13832t^10 13832*t^10 + 54340t^9 54340*t^9 + 157118t^8 157118*t^8 + 333580t^7 333580*t^7 + 509366t^6 509366*t^6 + 534820t^5 534820*t^5 + 354536t^4 354536*t^4 + 124852t^3 124852*t^3 + 15145t^2 15145*t^2 + 746t 746*t + 13)/t
}}}