gh-37226: unit group of number field: do not expand product when S is empty
    
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This PR fixes #36386, at least
for unit group computations (S-units will still suffer from the problem,
or when the units are even larger). This PR does **not** address
#31754, which would be a better
(but more complicated) solution for the problem.

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### ⌛ Dependencies
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URL: #37226
Reported by: AurelPage
Reviewer(s): AurelPage, Vincent Delecroix

Author: Release Manager

src/sage/rings/number_field/unit_group.py

@@ -305,6 +305,14 @@ def __init__(self, number_field, proof=True, S=None):
             sage: tuple(US(K(u)) for u in US.gens()) == US.gens()
             True
 
+        Bug :issue:`36386` (pari stack overflow while expanding units)::
+
+            sage: d = 12936642
+            sage: K = QuadraticField(d)
+            sage: K.unit_group(proof=False)
+            Unit group with structure C2 x Z of Number Field in a with defining polynomial x^2 - 12936642 with a = 3596.754370262167?
+
+
         """
         proof = get_flag(proof, "number_field")
         K = number_field
@@ -340,20 +348,20 @@ def __init__(self, number_field, proof=True, S=None):
         # compute the additional S-unit generators:
         if S:
             self.__S_unit_data = pK.bnfunits(pS)
+            # TODO: converting the factored matrix representation of bnfunits into polynomial
+            # form is a *big* waste of time
+            su = [pK.nfbasistoalg(pK.nffactorback(z)) for z in self.__S_unit_data[0][0:len(S)]]
+            su = [K(u, check=False) for u in su]
         else:
-            self.__S_unit_data = pK.bnfunits()
-        # TODO: converting the factored matrix representation of bnfunits into polynomial
-        # form is a *big* waste of time
-        su_fu_tu = [pK.nfbasistoalg(pK.nffactorback(z)) for z in self.__S_unit_data[0]]
-
-        self.__nfu = len(pK.bnf_get_fu())           # number of fundamental units
-        self.__nsu = len(su_fu_tu) - self.__nfu - 1 # number of S-units
-        self.__ntu = pK.bnf_get_tu()[0]             # order of torsion
+            su = []
+
+        self.__nfu = len(fu)            # number of fundamental units
+        self.__nsu = len(su)            # number of S-units
+        self.__ntu = pK.bnf_get_tu()[0] # order of torsion
         self.__rank = self.__nfu + self.__nsu
 
-        # Move the torsion unit first, then fundamental units then S-units
-        gens = [K(u, check=False) for u in su_fu_tu]
-        gens = [gens[-1]] + gens[self.__nsu:-1] + gens[:self.__nsu]
+        # Put the torsion unit first, then fundamental units then S-units
+        gens = [K(pK.bnf_get_tu()[1], check=False)] + fu + su
 
         # Construct the abstract group:
         gens_orders = tuple([ZZ(self.__ntu)]+[ZZ(0)]*(self.__rank))