This is a toy implementation used to generate examples for the preprint of Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication, as referenced in this page.

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

1. Clone the repository.
2. Edit sage script to point to the relevant directories.
   • DOT_SAGE is where the init.sage and hist.sage are expected to be found.
   • LOGFILEDIR is where you would like SAGE to save its log files.
   • RECIPDIR is where Marco Streng's RECIP installation is found.
   • FGAGDIR is where your finitely generated abelian group PARI/GP scripts are found.
   • SHIMURADIR is where your shimura.gp and shimura.macros.gp are found.
     • Note that shimura.gp is a script in Andreas Enge and Emmanuel Thomé's cmh. Its license is GPLv3 or later.
     • The shimura.gp in this repository is from an old version of the file. The rest of the script has not yet been tested on newer versions, but is expected to work.
   • Replace ~/.magma/2.24-3/ by where your magma installation lies.
3. Make sage executable.

1. Run .sage.
2. Load CMdata.sage via load("CMdata.sage")

This is an example usage to reproduce Example 4.11 as found in the preprint Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication.

1. A = ShGrpData([809, 53, 500],m=2)
2. pol = A.find_defining_polynomial_and_verify(rosenhain_invariants(2)[1],prec=400,verbose=True,check_conductor=True,autoretry=5)
3. pol[0] contains a polynomial that defines the extension $H_{K^r}(1) / K^r$.

Caveat: When using the polynomial to define an extension of (the SAGE object representing the) reflex field, SAGE attempts a polredbest, which is slow. I personally did not wait for SAGE to finish and instead used PARI/GP directly to verify. This is done by doing:

4. paripol = poly_s2p(pol[0], A.prelt['s2p'])
5. gp.rnfconductor(A.Kr_gp, [paripol, 10^6])

Verifying that the result is 1, we find that the conductor of the extension has no prime factors below $10^6$. If one is patient enough and wishes for PARI to find the conductor itself, replace [paripol, 10^6] with paripol.

• Documentation
• Handle cases when neither $\star_1$ nor $\star_2$ is true.