diff --git a/hpcgap/lib/basis.gd b/hpcgap/lib/basis.gd
index ee6297fc32..a8fb243ee5 100644
--- a/hpcgap/lib/basis.gd
+++ b/hpcgap/lib/basis.gd
@@ -786,9 +786,11 @@ DeclareOperation( "RelativeBasisNC", [ IsBasis, IsHomogeneousList ] );
 ##  (see&nbsp;<Ref Sect="Vector Spaces Handled By Nice Bases"/>).
 ##  <P/>
 ##  <A>name</A> must be a string,
-##  a filter <M>f</M> with this name is created, and
-##  a logical implication from <M>f</M> to <Ref Filt="IsHandledByNiceBasis"/>
-##  is installed.
+##  a filter <M>f</M> with this name is created which implies
+##  <Ref Filt="IsFreeLeftModule"/>, and
+##  a logical implication from the join of <M>f</M> with
+##  <Ref Filt="IsAttributeStoringRep"/> to
+##  <Ref Filt="IsHandledByNiceBasis"/> is installed.
 ##  <P/>
 ##  <A>record</A> must be a record with the following components.
 ##  <List>
@@ -805,28 +807,34 @@ DeclareOperation( "RelativeBasisNC", [ IsBasis, IsHomogeneousList ] );
 ##  </Item>
 ##  <Mark><C>NiceFreeLeftModuleInfo</C> </Mark>
 ##  <Item>
-##      the <C>NiceFreeLeftModuleInfo</C> method for left modules in <M>f</M>,
+##      the <Ref Attr="NiceFreeLeftModuleInfo"/> method for left modules in
+##      <M>f</M>,
 ##  </Item>
 ##  <Mark><C>NiceVector</C> </Mark>
 ##  <Item>
-##      the <C>NiceVector</C> method for left modules <M>V</M> in <M>f</M>;
+##      the <Ref Oper="NiceVector"/> method for left modules <M>V</M> in
+##      <M>f</M>;
 ##      called with <M>V</M> and a vector <M>v \in V</M>, this function returns the
 ##      nice vector <M>r</M> associated with <M>v</M>, and
 ##  </Item>
 ##  <Mark><C>UglyVector</C></Mark>
 ##  <Item>
-##      the <Ref Oper="UglyVector"/> method for left modules <M>V</M> in <M>f</M>;
-##      called with <M>V</M> and a vector <M>r</M> in the <C>NiceFreeLeftModule</C> value
-##      of <M>V</M>, this function returns the vector <M>v \in V</M> to which <M>r</M> is
+##      the <Ref Oper="UglyVector"/> method for left modules <M>V</M> in
+##      <M>f</M>;
+##      called with <M>V</M> and a vector <M>r</M> in the
+##      <Ref Attr="NiceFreeLeftModule"/> value of <M>V</M>,
+##      this function returns the vector <M>v \in V</M> to which <M>r</M> is
 ##      associated.
 ##  </Item>
 ##  </List>
 ##  <P/>
 ##  The idea is that all one has to do for implementing a new kind of free
 ##  left modules handled by the mechanism of nice bases is to call
-##  <C>DeclareHandlingByNiceBasis</C> and <C>InstallHandlingByNiceBasis</C>,
+##  <Ref Func="DeclareHandlingByNiceBasis"/> and
+##  <Ref Func="InstallHandlingByNiceBasis"/>,
 ##  which causes the installation of the necessary methods and adds the pair
-##  <M>[ f, </M><C><A>record</A>.detect</C><M> ]</M> to the global list <C>NiceBasisFiltersInfo</C>.
+##  <M>[ f, </M><A>record</A><C>.detect</C><M> ]</M> to the global list
+##  <Ref Var="NiceBasisFiltersInfo"/>.
 ##  The <Ref Oper="LeftModuleByGenerators"/> methods call
 ##  <Ref Func="CheckForHandlingByNiceBasis"/>, which sets the appropriate filter
 ##  for the desired left module if applicable.
@@ -889,6 +897,7 @@ InstallGlobalFunction( "DeclareHandlingByNiceBasis", function( name, info )
     local entry;
     atomic readwrite NiceBasisFiltersInfo do
     DeclareFilter( name );
+    InstallTrueMethod( IsFreeLeftModule, ValueGlobal( name ) );
     entry := [ ValueGlobal( name ), info ];
     MigrateObj( entry, NiceBasisFiltersInfo );
     Add( NiceBasisFiltersInfo, entry, 1 );
diff --git a/hpcgap/lib/basis.gi b/hpcgap/lib/basis.gi
index cf83dae265..89e4256d96 100644
--- a/hpcgap/lib/basis.gi
+++ b/hpcgap/lib/basis.gi
@@ -687,14 +687,16 @@ InstallGlobalFunction( "InstallHandlingByNiceBasis",
     filter:= ValueGlobal( name );
 
     # Install the detection of the filter.
+    # The mechanism is safe only if the domain can store
+    # its nice variant, thus we will install it only for cases where
+    # 'IsAttributeStoringRep' is guaranteed.
     atomic readwrite NiceBasisFiltersInfo do
       entry:= First( NiceBasisFiltersInfo,
                      x -> IsIdenticalObj( filter, x[1] ) );
       entry[3] := record.detect;
     od;
-
+    filter:= filter and IsAttributeStoringRep;
     InstallTrueMethod( IsHandledByNiceBasis, filter );
-    filter:= IsFreeLeftModule and filter;
 
     # Install the methods.
     InstallMethod( NiceFreeLeftModuleInfo,
@@ -1078,7 +1080,7 @@ InstallMethod( IsCanonicalBasis,
 ##
 #M  Basis( <V> )  . . . . . . . . . . . for free module handled by nice basis
 ##
-BasisForFreeModuleByNiceBasis:= function( V )
+BindGlobal( "BasisForFreeModuleByNiceBasis", function( V )
     local B;
     B:= Objectify( NewType( FamilyObj( V ),
                                 IsFiniteBasisDefault
@@ -1087,7 +1089,7 @@ BasisForFreeModuleByNiceBasis:= function( V )
                    rec() );
     SetUnderlyingLeftModule( B, V );
     return B;
-end;
+end );
 
 InstallMethod( Basis,
     "for free module that is handled by a nice basis",
diff --git a/hpcgap/lib/helpbase.gi b/hpcgap/lib/helpbase.gi
index 20ed39fce1..fe9de8ebe2 100644
--- a/hpcgap/lib/helpbase.gi
+++ b/hpcgap/lib/helpbase.gi
@@ -1442,4 +1442,3 @@ atomic readwrite HELP_REGION do
   fi;
 od; # end atomic
 end);
-
diff --git a/hpcgap/lib/polyconw.gi b/hpcgap/lib/polyconw.gi
index 51dc68134d..d1ac23a431 100644
--- a/hpcgap/lib/polyconw.gi
+++ b/hpcgap/lib/polyconw.gi
@@ -194,7 +194,7 @@ end);
 
 ##  list of all cases with less than 100*10^9 compatible polynomials, sorted
 ##  w.r.t. this number
-ConwayCandidates := function()
+BindGlobal( "ConwayCandidates", function()
   local cand, p, i;
   # read data
   for p in [2,113,1009] do
@@ -211,7 +211,7 @@ ConwayCandidates := function()
   Sort(cand);
   cand := Filtered(cand, a-> not IsBound(CONWAYPOLDATA[a[2]][a[3]]));
   return cand;
-end;
+end );
 
 ##
 ##
@@ -502,15 +502,15 @@ InstallGlobalFunction( IsCheapConwayPolynomial, function( p, n )
   fi;
   # this is not very precise, hopefully good enough for the moment
   if p < 41 then
-    if n < 100 and IsPrimeInt(n) then
+    if n < 100 and (n = 1 or IsPrimeInt(n)) then
       return true;
     fi;
   elif p < 100 then
-    if n < 40 and IsPrimeInt(n) then
+    if n < 40 and (n = 1 or IsPrimeInt(n)) then
       return true;
     fi;
   elif p < 1000 then
-    if n < 14 and IsPrimeInt(n) then
+    if n < 14 and (n = 1 or IsPrimeInt(n)) then
       return true;
     fi;
   elif p < 2^48 then
diff --git a/hpcgap/lib/ratfun.gi b/hpcgap/lib/ratfun.gi
index 6fa6d08fcb..ce3f2ad67c 100644
--- a/hpcgap/lib/ratfun.gi
+++ b/hpcgap/lib/ratfun.gi
@@ -1078,6 +1078,20 @@ function( left, right )
   return SumCoefPolynomial(left, right);
 end);
 
+# divide by constant polynomials
+
+InstallMethod(\/,"constant denominator poly",IsIdenticalObj,
+  [IsPolynomial,IsPolynomial],0,
+function(num,den)
+local e;
+  e:=ExtRepNumeratorRatFun(den);
+  if Length(e)=0 then Error("Division by zero");
+  elif Length(e)>2 or Length(e[1])>0 then TryNextMethod();fi;
+  e:=Inverse(e[2]);
+  if e=fail then TryNextMethod();fi;
+  return e*num;
+end);
+
 InstallGlobalFunction( QuotientPolynomialsExtRep,QUOTIENT_POLYNOMIALS_EXT);
 
 #############################################################################
@@ -1090,9 +1104,7 @@ InstallMethod(HeuristicCancelPolynomialsExtRep,"ignore",true,
   [IsRationalFunctionsFamily,IsList,IsList],
   # fallback: lower than default for the weakest conditions
   -1,
-function(f,a,b)
-  return fail; # can't do anything
-end);
+ReturnFail);
 
 InstallGlobalFunction(TryGcdCancelExtRepPolynomials,TRY_GCD_CANCEL_EXTREP_POL);
 
@@ -1129,7 +1141,7 @@ local fam,t,num,tt,den;
                   ExtRepPolynomialRatFun(left),
                   ExtRepDenominatorRatFun(right));
       num:=ZippedProduct(t[1],ExtRepNumeratorRatFun(right),
-            fam!.zeroCoefficient,fam!.zippedProduct);
+          fam!.zeroCoefficient,fam!.zippedProduct);
       if Length(t[2])=2 and t[2][1]=[] and t[2][2]=fam!.oneCoefficient then
           return PolynomialByExtRepNC(fam,num);
       else
@@ -1140,7 +1152,7 @@ local fam,t,num,tt,den;
                   ExtRepPolynomialRatFun(right),
                   ExtRepDenominatorRatFun(left));
       num:=ZippedProduct(t[1],ExtRepNumeratorRatFun(left),
-            fam!.zeroCoefficient,fam!.zippedProduct);
+          fam!.zeroCoefficient,fam!.zippedProduct);
       if Length(t[2])=2 and t[2][1]=[] and t[2][2]=fam!.oneCoefficient then
           return PolynomialByExtRepNC(fam,num);
       else
@@ -1152,9 +1164,9 @@ local fam,t,num,tt,den;
       tt:=TryGcdCancelExtRepPolynomials(fam,
           ExtRepNumeratorRatFun(right),ExtRepDenominatorRatFun(left));
       num:=ZippedProduct(t[1],tt[1],fam!.zeroCoefficient,
-                    fam!.zippedProduct);
+          fam!.zippedProduct);
       den:=ZippedProduct(t[2],tt[2],fam!.zeroCoefficient,
-                    fam!.zippedProduct);
+          fam!.zippedProduct);
       if Length(den)=2 and den[1]=[] and den[2]=fam!.oneCoefficient then
           return PolynomialByExtRepNC(fam,num);
       else
@@ -1219,14 +1231,14 @@ local fam,num,den,lnum,rnum,lden,rden,t,tmp,tmpp,i;
       # same denominator: add numerators
       num:=ZippedSum(lnum,rnum,fam!.zeroCoefficient,fam!.zippedSum);
       if Length(num)=0 then
-          return Zero(fam);
+        return Zero(fam);
       fi;
       t:=TryGcdCancelExtRepPolynomials(fam,num,lden);
       if Length(t[2])=2 and Length(t[2][1])=0 and t[2][2]=fam!.oneCoefficient
         then
-          return PolynomialByExtRepNC(fam,t[1]);
+        return PolynomialByExtRepNC(fam,t[1]);
       else
-          return RationalFunctionByExtRepNC(fam,t[1],t[2]);
+        return RationalFunctionByExtRepNC(fam,t[1],t[2]);
       fi;
     else
       t:=TryGcdCancelExtRepPolynomials(fam,lden,rden);
@@ -1236,9 +1248,9 @@ local fam,num,den,lnum,rnum,lden,rden,t,tmp,tmpp,i;
                     fam!.zippedProduct);
       num:=ZippedSum(tmp,tmpp,fam!.zeroCoefficient,fam!.zippedSum);
       if Length(t)=3 then
-          tmp:=t[3];
+        tmp:=t[3];
       else
-          tmp:=QuotientPolynomialsExtRep(fam,rden,t[2]);
+        tmp:=QuotientPolynomialsExtRep(fam,rden,t[2]);
       fi;
       tmpp:=TryGcdCancelExtRepPolynomials(fam,num,tmp);
       den:=ZippedProduct(tmpp[2],t[1],fam!.zeroCoefficient,