fix: remove obsolete limitation in torsion unit computation (#1822)

Author: thofma

src/NumFieldOrd/NfOrd/TorsionUnits.jl

@@ -251,6 +251,8 @@ end
 #
 # Magma code:
 # [Maximum(&cat[ EulerPhiInverse(d) : d in Divisors(n)  | #EulerPhiInverse(d) ne 0 ]) : n in [1..250]];
+# Hecke
+# [ maximum(vcat([euler_phi_inv(d) for d in divisors(n)]...)) for n in 1:250]
 const _euler_phi_inverse_maximum =
 [ 2, 6, 2, 12, 2, 18, 2, 30, 2, 22, 2, 42, 2, 6, 2, 60, 2, 54, 2, 66, 2, 46, 2,
 90, 2, 6, 2, 58, 2, 62, 2, 120, 2, 6, 2, 126, 2, 6, 2, 150, 2, 98, 2, 138, 2,
@@ -265,14 +267,19 @@ const _euler_phi_inverse_maximum =
 2, 6, 2, 810, 2, 6, 2, 726, 2, 446, 2, 870, 2, 454, 2, 458, 2, 94, 2, 708, 2,
 158, 2, 12, 2, 478, 2, 1050, 2, 46, 2, 12, 2, 166, 2, 30, 2, 502 ]
 
+#
+
 # One should/could also try to be closer to Algorithm 1
 # in Molin, "On the calculation of roots of unity in a number field"
 function _torsion_group_order_divisor(K::AbsSimpleNumField)
-
   if degree(K) <= 250
     upper_bound = _euler_phi_inverse_maximum[degree(K)]
   else
-    error("Not implemented yet")
+    l = 0
+    for d in divisors(degree(K))
+      l = maximum(euler_phi_inv(d); init = l)
+    end
+    upper_bound = l
   end
 
   p = upper_bound + 1
@@ -342,7 +349,11 @@ function _torsion_group_order_divisor(K::NumField)
   if absolute_degree(K) <= 250
     upper_bound = _euler_phi_inverse_maximum[absolute_degree(K)]
   else
-    error("Not implemented yet")
+    l = 0
+    for d in divisors(absolute_degree(K))
+      l = maximum(euler_phi_inv(d); init = l)
+    end
+    upper_bound = l
   end
 
   p = upper_bound + 1

test/NfOrd/TorsionUnits.jl

@@ -19,4 +19,11 @@
   K, a = cyclotomic_field(13, cached = false)
   O = maximal_order(K)
   @test (@inferred length(Hecke._torsion_units_lattice_enum(O)[1])) == 26
+
+  let
+    p = 2^9
+    K, a = cyclotomic_field(p)
+    l = Hecke._torsion_group_order_divisor(K)
+    @test is_divisible_by(degree(K), euler_phi(l))
+  end
 end