The OSCAR Book Companion
The OSCAR Book Companion
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 defined mod (<3, 3>, InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}[]) of 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