**********************************************************************
* The data in -small.db- are
* Copyright (C) 2008 Nils-Peter Skoruppa
*
* Distributed under the terms of the GNU General Public License (GPL)
* http://www.gnu.org/licenses/
**********************************************************************
Author: Nils-Peter Skoruppa
File Type: sqlite database
Remarks:
The data were computed on the Rubens cluster of the university of
Siegen using PARI/GP and the ext72 collection of PARI/GP scripts
written for this purpose by the author. A detailed description of the
computations will be published elsewhere.
Description:
The minus part $C^-$ of the ideal class group $C$ of
$K=\Q(\zeta_{71})$ is cyclic of order $49*79241=3 882 809$. (The
plus-part $C_+$ is trivial.) The action of the Galois group
$G=Galois(K)$ thus defines a group homomorphism $G \mapsto Aut(C) =
(Z/hZ)^*$. In fact, this is an injection and the image of this map is
the subgroup $$ of $Aut(C)$ generated by the residue class
mod $h$ of $w=1137674$. As representative for a class $X$ of $G\C = \((Z/hZ)$ we take $min(X)$, the minimal non-negative $n$ such
that $n+hZ$ lies in the orbit $X$. Let $R$ be the set of these
representatives without $0$. The order of $R$ is $55474$.
Let $J$ be an ideal of $K$. Associated to $J$ we have a (equivalence
class of) {\it small} even lattice(s) of rank 70, determinant 71, and
having an automorphism of order 71, and a (equivalence class of a)
{\it big} lattice, i.e. an even unimodular lattice of rank $72$ with
automorphism of order $71$. These maps factor through an isomorphism
of $G\C$ onto the set of these classes of lattices, respectively.
Let $p$ be a prime ideal dividing $569$. The database big.db contains
in its (single) table small for each $n \in R$ a Gram matrix $G$ of
the small lattice class associated to $p^n$, and a square matrix $M$
representing the automorphism of oder $71$ of the lattice defined by
$G$ (i.e. $M^tGM=G$ and $order(M)=71$). The rows of the table small
are
n gram_m auto_m
The content of auto_m is the sequence of numbers obtained by
concatenating the rows of $M$, the content of the symmetric matrix
gram_s is the sequence obtained by concatenating the first part of
each row (up to and including the respective diagonal element).
(C) Countnumber, 2007