This is a code for class group computation for imaginary multiquadradic fields.

It is built on top of the implementation of the Biasse-van Vredendaal algorithm for the class group computation for real multiquadratic fields:

• https://scarecryptow.org/publications/multiclass.html

Requirements

• Sage 9.4+

How to compile

1. Run make
2. Edit paths in Makefile according to your installation if code above fails.

Folder structure

• tests/ - unit tests
• trees/ - trees describing prime ideal splitting over subfields. Factor base is loaded from these files. They are generated by trees_generation.sage script.
• relations/ - class group relation matrices for every subfield. They are generated by testrelations.sage script.
• experiments/ - examples of class group computation.

How to use

1. Set the target field in the file trees_generation.sage (food variable). Alternatively, you can pass d_1 d_2 ... d_n as command line arguments of this script.
2. Run sage trees_generation.sage to generate trees.
3. Run sage testrelations.sage to compute class group and relation matrices for subfields.
4. Run sage clgp_verify.sage d_1 d_2 ... d_n --log-zeta-res z --class-number h to verify result of computation. Here z is the residue of Dedekind zeta function at 1. This value can be computed using Hecke, see examples in ''experiments'' folder.

Limitations

• Currently only prime $d_1, ..., d_n$ where $d_i \equiv 1 \bmod 4$ are supported. The output for $d_i \equiv 3 \bmod 4$ contains extra factors and thus partial.