These files can be read directly into gp. Each file contains a single
gp list whose entries correspond to a single local field of over \Q_p
of the specified degree.
Each field is itself represented by a list. We will describe many of
the entries here. Some entries are now obsolete, or are used while
producing the database. Others are redundant, but are included to
make it easier to sort the list in a variety of ways.
1. the prime p
2. c, where the discriminant of the field is (p^c)
3. e, the ramification index
4. f, the residue field degree
5. n, the degree of the extension
6. our defining polynomial
7. Galois group
8. the order of the Galois group
9. the inertia subgroup
10. a list of wild slopes
11. the pair [t,u] where t and u are the degrees of the tame ramified
and unramified degrees of the Galois closure
12. a pair for defining the degree n field over \Q_p, first our
defining polynomial for the maximal unramified subextension K^u,
then an Eisenstein polynomial for our field over K^u. This entry
is used in applying Panayi's root finding algorithm.
13. list of fields to replace with when computing grd's of composita.
This may be just 0 meaning to keep the current polynomial.
14. the discriminant root field given as a pair, usually [1 or p, 0 or
1]. A second coordinate of 1 means to multiply the first
coordinate by a non-square modulo p. For p=2, the first
coordinate can be +/- 1, +/- 2, and the second coordinate signals
whether or not to take an unramified twist.
15. first coordinate of entry 14, normalized for nicer sorting. If
p=2 and the entry is <0, it gets multiplied by -3/2 so that original
values get sorted 1, -1, 2, -2.
16. second coordinate of entry 14, (here again for sorting)
17. root number
18. subfields of the degree n field as a list of pairs, [degree of
subfield over \Q_p, list of defining polynomials]. This could
look like [[2, []], [3, [x^3 - x + 1]]] to indicate that 2
divides the degree, but there are no quadratic subfields, and
there is one cubic subfield.
19. holding spot for intermediate calculations
20. factors of twin algebra
21. Galois mean slope
22. number of automorphisms of the degree n field
23. another holding spot for intermediate calculations
Galois/inertia groups are coded as a 3-tuple or 4-tuple (i.e., the
4th coordinate is optional) of the form
[group order, type, data, html for displaying]
The type is one of
"t": the group is given by "T" information, the data is [n, t]
"i": the group sits intransitively in S_n. If data is given, it
is a [n,t] isomorphic to the group.
"e": it is the trivial group
"a": abstract group, so no T-number. 4th coordinate must be present
to know how to display this group.
"o": just the order of the group is known
"tu": we have extension information, so the group is C_t.C_u. The
data contains [t,u]