The tables are organized by the degree over
Qp and contain the following data:
c
= p-adic valuation of the discriminant
e
= ramification index
f
= residue field degree
d
represents the discriminant root field,
Qp(sqrt(disc(f)), where f is
any irreducible defining polynomial for the field. This field can be
taken in the form
Qp(sqrt(d)) where we pick representatives
d in Qp× modulo
squares. Here, * = 5 if p = 2 and
* is a quadratic non-residue if p is odd.
Thus for p odd, the possibilities for d are
1, *, p, and p*, and for p=2,
d is one of 1, *, 2, 2*, -1, -*, -2, or -2*.
= local root number
Polynomial
= a sample defining polynomial. The polynomial is a link to
the same polynomial in a form which can be readily selected
and pasted into other programs.
G
= Galois group, given in many cases
in descriptive notation such as Cn for the cyclic group
of order n, otherwise via its T number as in, for example,
"The transitive groups of
degree up to eleven", by Butler and McKay, Comm. Algebra 11, 1983.
In all cases, Galois group names are links to pop-ups with group information.
I = Inertia group. These follow the same
conventions as for Galois groups. In particular, they
are clickable links to further information. In some cases, the
inertia subgroup entry may only give partial information, such as the
order of the inertia subgroup.
Slope Content gives information about the higher ramification
filtration for the Galois group G, using Artin's upper
numbering. It is explained in detail in the paper at the top
of this page. In short, slope content takes the form
[s_1, s_2, ..., s_j]^u_t.
The s_i give indices where the wild higher ramification
groups change counted with multiplicity, so that a value is repeated
m times if the order of the corresponding ramification
group changes by a factor of pm. If there
is no wild ramification, then no values are printed between
the brackets. The values
t and u give the tamely ramified and unramified degrees
of the Galois closure respectively. We do not print
t or u if its value is 1.
GMS =
Galois Mean Slope, equal to the exponent of p in the
root discriminant of the Galois closure of the field
Deg-j Subs = list of subfields of degree j over
Qp. Quadratic subfields are coded as
listed above. Other subfields are given by their sample
polynomials, with the exception of unramified subfields.
Unramified degree n subfields are denoted simply by Un.
Twin Algebra
= a defining polynomial for the twin algebra of a sextic
field. This twin polynomial is factored as a product of
sample polynomials.