The OSCAR Book Companion
The OSCAR Book Companion
This page contains the code samples from the corresponding chapter in the OSCAR book. You can access the full chapter here.
julia> ch = character_table("Co2", 2)[2];
julia> degree(ch)
22
julia> Oscar.OrthogonalDiscriminants.od_from_order(ch)
(true, "O+")
julia> ch = character_table("Co3", 3)[2];
julia> degree(ch)
22
julia> Oscar.OrthogonalDiscriminants.od_from_eigenvalues(ch)
(true, "O+")
julia> ch = character_table("A12")[26];
julia> degree(ch)
1728
julia> Oscar.OrthogonalDiscriminants.od_for_specht_module(ch)
(true, "1")
julia> ch = character_table("R(27)")[16];
julia> degree(ch)
18278
julia> Oscar.OrthogonalDiscriminants.od_from_p_subgroup(ch, 3)
(true, "-3")
julia> Oscar.OrthogonalDiscriminants.show_with_ODs(
character_table("G2(3)", 2))
G2(3)mod2
2 6 3 3 . 1 1 . . . . . .
3 6 6 6 6 4 4 . 3 3 3 . .
7 1 . . . . . 1 . . . . .
13 1 . . . . . . . . . 1 1
1a 3a 3b 3c 3d 3e 7a 9a 9b 9c 13a 13b
2P 1a 3a 3b 3c 3d 3e 7a 9a 9c 9b 13b 13a
3P 1a 1a 1a 1a 1a 1a 7a 3c 3c 3c 13a 13b
7P 1a 3a 3b 3c 3d 3e 1a 9a 9b 9c 13b 13a
13P 1a 3a 3b 3c 3d 3e 7a 9a 9b 9c 1a 1a
d OD 2
X_1 1 + 1 1 1 1 1 1 1 1 1 1 1 1
X_2 1 O- + 14 5 5 -4 2 -1 . 2 -1 -1 1 1
X_3 2 o 64 -8 -8 1 4 -2 1 1 A /A -1 -1
X_4 2 o 64 -8 -8 1 4 -2 1 1 /A A -1 -1
X_5 1 O- + 78 -3 -3 -3 -3 6 1 . . . . .
X_6 1 O+ + 90 9 9 9 . . -1 . . . -1 -1
X_7 1 O- + 90 -9 18 . 3 -3 -1 -3 . . -1 -1
X_8 1 O- + 90 18 -9 . 3 -3 -1 -3 . . -1 -1
X_9 1 O- + 378 -9 -9 9 -3 -6 . 3 . . 1 1
X_10 2 O+ + 448 16 16 -11 -2 -2 . 1 1 1 B B*
X_11 2 O+ + 448 16 16 -11 -2 -2 . 1 1 1 B* B
X_12 1 O+ + 832 -32 -32 -5 4 4 -1 1 1 1 . .
A = 3z_3 + 1
/A = -3z_3 - 2
B = -z_13^11 - z_13^8 - z_13^7 - z_13^6 - z_13^5 - z_13^2 - 1
B* = z_13^11 + z_13^8 + z_13^7 + z_13^6 + z_13^5 + z_13^2
julia> show_OD_info("G2(3)")
G2(3): 2^6*3^6*7*13
--------------------
i| chi| K|disc| 2|3| 7| 13
--+----+------+----+------------+-+-------------+--------
2| 14a| Q| -3| 14a| | 14a| 14a
| | | | O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
5| 78a| Q| -3| 78a| | 78a| 78a
| | | | O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
9|104a| Q| 21| 14a+90a| | (def. 1)| 104a
| | | | O-, O+| | | O-
--+----+------+----+------------+-+-------------+--------
10|168a| Q| 13| 78a+90a| | 168a|(def. 1)
| | | | O-, O+| | O-|
--+----+------+----+------------+-+-------------+--------
11|182a| Q| -3| 14a+78a+90c| | 182a| 182a
| | | | O-, O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
12|182b| Q| -3| 14a+78a+90b| | 182b| 182b
| | | | O-, O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
15|448a|Q(b13)| 1| 448a| | 448a|14a+434a
| | | | O+| | O+| O+, O+
--+----+------+----+------------+-+-------------+--------
16|448b|Q(b13)| 1| 448b| | 448b|14a+434a
| | | | O+| | O+| O+, O+
--+----+------+----+------------+-+-------------+--------
17|546a| Q| -3|78a+90b+378a| | 546a| 546a
| | | | O-, O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
18|546b| Q| -3|78a+90c+378a| | 546b| 546b
| | | | O-, O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
19|728a| Q| 1| 14a+378a| | 728a| 728a
| | | | O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
20|728b| Q| 1| 14a+378a| | 728b| 728b
| | | | O-, O-| | O+| O+
--+----+------+----+------------+-+-------------+--------
23|832a| Q| 1| 832a| |64ab+78a+626a| 832a
| | | | O+| | O+, O+, O+| O+
julia> K, _ = quadratic_field(13)
(Real quadratic field defined by x^2 - 13, sqrt(13))
julia> ray_class_field(3*maximal_order(K))
Class field
over real quadratic field defined by x^2 - 13
with modulus
finite part <3>
infinite part
[]
with structure
Z/1
julia> function is_galois_discriminant_field(data)
chi = Oscar.OrthogonalDiscriminants.character_of_entry(data)
F, emb = character_field(chi)
c = conductor(emb(gen(F)))
galgens = Oscar.AbelianClosure.generators_galois_group_cyclotomic_field(c)
delta = atlas_irrationality(data[:valuestring])
return all(x -> is_square(preimage(emb, delta * x(delta))),
galgens)
end;
julia> info = all_od_infos(characteristic => 0, is_simple);
julia> filter!(r -> r[:valuestring] != "?" &&
conductor(atlas_irrationality(r[:valuestring])) > 1,
info);
julia> length(info)
58
julia> filter!(!is_galois_discriminant_field, info);
julia> length(info)
26
julia> println(sort!(collect(Set([r[:groupname] for r in info]))))
["HN", "He", "J1", "J3", "ON", "Ru"]
julia> info = all_od_infos(characteristic => 0, degree => 1);
julia> all(x -> x[:valuestring] == "?" ||
is_odd(parse(Int, x[:valuestring])),
info)
true
julia> plus = []; minus = [];
julia> for d in all_od_infos()
if d[:valuestring] == "O+"
push!(plus, (d[:groupname], d[:characteristic], d[:degree],
parse(Int, filter(isdigit, d[:charname]))))
elseif d[:valuestring] == "O-"
push!(minus, (d[:groupname], d[:characteristic], d[:degree],
parse(Int, filter(isdigit, d[:charname]))))
end
end
julia> both = intersect!(plus, minus);
julia> filter(x -> x[2] == 2, both)
1-element Vector{Any}:
("G2(3)", 2, 1, 90)
julia> length(both)
103