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Finding Hauptmoduls

asked 2011-06-29 19:10:45 -0500

anonymous user

Anonymous

Given a modular curve with genus zero (for example $X_{1}(7)$) is there a way in SAGE to find its Hauptmodul?

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answered 2011-08-18 06:47:53 -0500

John Cremona gravatar image

updated 2011-08-18 07:51:47 -0500

niles gravatar image

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 + 36*t^3 + 270*t^2 + 756*t + 729)/t
sage: Fricke_modules[5]
(t^6 + 30*t^5 + 315*t^4 + 1300*t^3 + 1575*t^2 + 750*t + 125)/t
sage: Fricke_modules[7]
(t^8 + 28*t^7 + 322*t^6 + 1904*t^5 + 5915*t^4 + 8624*t^3 + 4018*t^2 + 748*t + 49)/t
sage: Fricke_modules[13]
(t^14 + 26*t^13 + 325*t^12 + 2548*t^11 + 13832*t^10 + 54340*t^9 + 157118*t^8 + 333580*t^7 + 509366*t^6 + 534820*t^5 + 354536*t^4 + 124852*t^3 + 15145*t^2 + 746*t + 13)/t
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Comments

Instead of the {{{ }}}'s try indenting the code like a python codeblock or select the code and click the button in the toolbar with the 0101010's.

benjaminfjones gravatar imagebenjaminfjones ( 2011-08-18 06:49:39 -0500 )edit
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answered 2011-08-19 03:06:43 -0500

John Cremona gravatar image

Here's a followup mainly to test if I can get the hang of the code entry: The curve $X_{split}(5)$ has genus $0$ and degree $15$, and a Hauptmodul is $s$ where $$j=\frac{(s+5)^3(s^2-5)^3(s^2+5s+10)^3}{(s^2+5s+5)^5}.$$

Secondly the Hauptmoduln for all genus zero $X_0(N)$ are given in the very nice paper by Robert Maier (2008) http://arxiv.org/abs/math/0611041. In fact we derived the one for $X_{split}(5)$ from Maier's for $X_0(25)$.

OK, so there was no code entry after all, but I did get to include a link!

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Asked: 2011-06-29 19:10:45 -0500

Seen: 693 times

Last updated: Aug 19 '11