feat: improve locally free class group (thofma#1573)

- compute the idempotents in the center

src/AlgAssAbsOrd/LocallyFreeClassGroup.jl

@@ -233,7 +233,22 @@ function K1_order_mod_conductor(O::AlgAssAbsOrd, OA::AlgAssAbsOrd, F::AlgAssAbsO
   end
   # Make the generators coprime to the other ideals
   if length(moduli) != 0 # maybe O is maximal
-    k1 = make_coprime(elements_for_crt, moduli)
+    # we compute the idempotents using the ideals in the center
+    # and map them to the order
+    # this is faster than computing the idempotents in the order itself
+    if length(moduli) >= 2
+      idemZ = _idempotents_for_make_coprime([q * OinZ for (q,_) in prim])
+      idems = [O(ZtoA(elem_in_algebra.(x))) for x in idemZ]
+      for i in 1:length(idems)
+        u = idems[i]
+        @assert u in moduli[i]
+        @assert all(1 - u in moduli[j] for j in 1:length(moduli) if j != i)
+      end
+    else
+      # only one moduli, so no idempotent
+      idems = elem_type(O)[]
+    end
+    k1 = make_coprime(elements_for_crt, moduli, idems)
   else
     k1 = elem_type(O)[]
   end

src/NumFieldOrd/NfOrd/ResidueRingMultGrp.jl

@@ -1170,9 +1170,20 @@ end
 #
 ################################################################################
 
+function _idempotents_for_make_coprime(ideals)
+  products = _compute_products_for_make_coprime(ideals)
+  n = length(ideals)
+  res = elem_type(order_type(algebra(ideals[1])))[]
+  for i = 1:n
+    u, v = idempotents(ideals[i], products[i])
+    push!(res, u)
+  end
+  return res
+end
+
 # For an element x of elements[i] this computes an element y with
 # x \equiv y mod ideals[i] and x \equiv 1 mod ideals[j] for all j not equal i.
-function make_coprime(elements::Vector{Vector{S}}, ideals::Vector{T}) where { S <: Union{ AbsNumFieldOrderElem, AlgAssAbsOrdElem }, T <: Union{ AbsNumFieldOrderIdeal, AlgAssAbsOrdIdl } }
+function make_coprime(elements::Vector{Vector{S}}, ideals::Vector{T}, idempotents::Vector{S}) where { S <: Union{ AbsNumFieldOrderElem, AlgAssAbsOrdElem }, T <: Union{ AbsNumFieldOrderIdeal, AlgAssAbsOrdIdl } }
   @assert !isempty(ideals)
   @assert length(elements) == length(ideals)
 
@@ -1181,20 +1192,33 @@ function make_coprime(elements::Vector{Vector{S}}, ideals::Vector{T}) where { S
     return elements[1]
   end
 
-  products = _compute_products_for_make_coprime(ideals)
-
   One = one(order(ideals[1]))
   result = Vector{S}()
   for i = 1:n
-    u, v = idempotents(ideals[i], products[i])
+    u = idempotents[i]
+
     for j = 1:length(elements[i])
-      t = elements[i][j] * v + One * u
+      t = elements[i][j] * (1 - u) + One * u
       push!(result, t)
     end
   end
   return result
 end
 
+function make_coprime(elements::Vector{Vector{S}}, ideals::Vector{T}) where { S <: Union{ AbsNumFieldOrderElem, AlgAssAbsOrdElem }, T <: Union{ AbsNumFieldOrderIdeal, AlgAssAbsOrdIdl } }
+  @assert !isempty(ideals)
+  @assert length(elements) == length(ideals)
+
+  n = length(ideals)
+  if n == 1
+    return elements[1]
+  end
+
+  idempotents = _idempotents_for_make_coprime(ideals)
+
+  return make_coprime(elements, ideals, idempotents)
+end
+
 # Build the products \prod_{j\neq i} ideals[j] for all i
 function _compute_products_for_make_coprime(ideals::Vector{T}) where { T <: Union{ AbsNumFieldOrderIdeal, AlgAssAbsOrdIdl } }
   n = length(ideals)