Class groups of quadratic imaginary fields

Data from extensive computations on class groups of quadratic imaginary fields is available below. It is organized by fundamental discriminant $-d$, and divided into four groups based on congruences:

• $d\equiv 3\pmod 8$
• $d\equiv 7\pmod 8$
• $d\equiv 4\pmod {16}$
• $d\equiv 8\pmod {16}$

For each congruence class above, there are 4096 files, indexed from $k=0$ to $k=4095$. The $k$th file contains data for $k\cdot 2^{28} \leq |d| \lt (k+1)\cdot 2^{28}$. The files are named accordingly, so the $k=12$th file for $d\equiv 7\pmod8$ is called cl7mod8.12.gz, the final extension because it has been compressed with gzip. The compressed files range in size from 50 to 200 megabytes.

File and data format

The $k$th file of data for $d\equiv r\pmod{m}$ has filename of the form cl{$r$}mod{$m$}.{$k$} after uncompressing with gzip (for example, file $12$ for $d\equiv7\pmod8$ is cl7mod8.12). In this file there is one line per discriminant, with discriminants in order of their absolute value. The discriminants and associated class group data may be extracted as follows, where for $i\ge1$ we define $-d_i$ to be the $i$th discriminant of the file:

• Initialise $-d_0=k\cdot2^{28}+r$.
• For $i\ge1$, let the data in line $i$ of the file be

  $a$

  $b$

  $c_1\ c_2\ \ldots\ c_t$

• Then
  • $d_{i} = d_{i-1}+m\cdot a$,
  • $h(-d_{i}) = b$,
  • the invariant factors for the class group are $[c_1, c_2,\ldots, c_t]$.
  In particular, $b=\prod_{j=1}^t c_j$.

For example, the first two lines of file cl4mod16.1 are

$0$	$12160$	$380\ 4\ 4\ 2$
$2$	$4392$	$2196\ 2$

so

• $-d_0=1\cdot2^{28}+4=268435460$, and then
• $-d_1=-d_0+16\cdot0=268435460$, with class number $12160$,
• $-d_2=-d_1+16\cdot2=268435492$, with class number $4392$,

and so on.

File download

$d \equiv $
$k = $	integer from 0 to 4095