Looking for a database of results in number theory Is there a public database in this world consisting of known number fields, their discriminants, and their ideal class groups, etc? If so, how does a lay person like me have access to this database?
 A: Hideo Wada, A table of ideal class groups of imaginary quadratic fields, Proc. Japan Acad. Volume 46, Number 5 (1970), 401-403 is available at Link.
I found that by typing $$\rm table\ class\ group$$ into Google, and I expect more can be found by that technique.
A: From PARI's tables: This directory contains a stripped down version of the number field tables
published by the Bordeaux computational number theory group (H. Cohen,
F. Diaz y Diaz, M. Olivier and their students) around 1995, and incorporates
a number of corrections. The original [uncorrected] tables can be found at ftp://megrez.math.u-bordeaux.fr/pub/numberfields/
AUTHORS: See ftp://megrez.math.u-bordeaux.fr/pub/numberfields/readme.pdf
for the authors of the original tables (~1995). The present version was
set up by Karim Belabas (Bordeaux) for the PARI group (2007).
FORMAT: The file T.gp contains data pertaining to fields of degree
$\rm\:3 \le n \le 7\:$ with $\rm\:0 \le r \le n\:$ real places. The data is in GP format, one
field per line, meant to be fed to the gp calculator as in
v = readvec("T31.gp");
which stores in the vector $\rm\: v\ (182417$ elements) data corresponding to the
$182417$ complex cubic fields with discriminant $\:\!> -10^6.$ Reading the largest
table requires a PARI stack size around $\rm\:\!30\:\!M$.
Entries in the resulting vector are sorted by increasing discriminant absolute
value. Each entry is a $4$-components vector: [disc, V, h, cyc], where


*

*disc is the field discriminant.

*V is the VECTOR of coefficients of the minimal polynomial of a generator
of the number field. P = Pol(V) is the true defining polynomial, and
bnfinit(P) will recompute the bnf data associated to the field.
polgalois(P) indicates the Galois group of its Galois closure.  

*h is the class number.

*cyc is the vector of orders of the class group cyclic components in
"elementary divisors" form: cyc $\rm = [d_1, d_2, \ldots, d_k]$ means that
 $\rm\:Cl(K) \cong (\mathbb Z/d_1 \mathbb Z) \times\cdots \times (\mathbb Z/d_k \mathbb Z),$  with $\rm\:d_k\: |\: \cdots\:|\: d_2\: |\: d_1.$

A: The L-functions and modular forms database (LMFDB) has an extensive and searchable database of number fields, including all the information you mentioned.
