The Latin Modern fonts are derived from the famous Computer Modern fonts designed by Donald E. Knuth and described in Volume E of his Computers & Typesetting series.
This fonts are authors' property, and are either shareware, demo versions or public domain. The licence mentioned above the download button is just an indication. Please look at the readme-files in the archives or check the indicated author's website for details, and contact him if in doubt. If no author/licence is indicated that's because we don't have information, that doesn't mean it's free.
% This is version 1.0, dated 22 June 2009, of the GUST Font License.
% (GUST is the Polish TeX Users Group, http://www.gust.org.pl)
%
% For the most recent version of this license see
% http://www.gust.org.pl/fonts/licenses/GUST-FONT-LICENSE.txt
% or
% http://tug.org/fonts/licenses/GUST-FONT-LICENSE.txt
%
% This work may be distributed and/or modified under the conditions
% of the LaTeX Project Public License, either version 1.3c of this
% license or (at your option) any later version.
%
% Please also observe the following clause:
% 1) it is requested, but not legally required, that derived works be
% distributed only after changing the names of the fonts comprising this
% work and given in an accompanying "manifest", and that the
% files comprising the Work, as listed in the manifest, also be given
% new names. Any exceptions to this request are also given in the
% manifest.
%
% We recommend the manifest be given in a separate file named
% MANIFEST-
% of the font family. If a separate "readme" file accompanies the Work,
% we recommend a name of the form README-
%
% The latest version of the LaTeX Project Public License is in
% http://www.latex-project.org/lppl.txt and version 1.3c or later
% is part of all distributions of LaTeX version 2006/05/20 or later.
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