diff --git a/drafty/draft.sage b/drafty/draft.sage
index ad3236f..936c85f 100644
--- a/drafty/draft.sage
+++ b/drafty/draft.sage
@@ -1,7 +1,7 @@
-p = 3
-F = GF(p^2, 'a')
+p = 5
+F = GF(p).algebraic_closure('t')
 #F1 = F.algebraic_closure('t')
-a = F.gens()[0]
+a = F.gen(2)
 R.<x> = PolynomialRing(F)
 P1 = superelliptic(x, 1)
-AS = as_cover(P1, [P1.x^2, a*P1.x^2])
+AS = as_cover(P1, [P1.x^2, a*P1.x^2])
\ No newline at end of file
diff --git a/superelliptic/superelliptic_class.sage b/superelliptic/superelliptic_class.sage
index 0664f79..c99a16c 100644
--- a/superelliptic/superelliptic_class.sage
+++ b/superelliptic/superelliptic_class.sage
@@ -238,38 +238,34 @@ class superelliptic:
             b += M
         return (C.x)^a/(C.y)^b
 
-def reduction(curve, g):
+def reduction(C, g):
     '''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
     it replaces repeteadly all y^m's in g(x, y) by f(x). As a result
     you obtain sum_{i = 0}^{m-1} y^i g_i(x).'''
-    p = curve.characteristic
-    F = curve.base_ring
-    Fxy, Rxy, x, y = curve.fct_field
-    f = curve.polynomial
+    p = C.characteristic
+    F = C.base_ring
+    Rxy.<x, y> = PolynomialRing(F, 2)
+    Fxy = FractionField(Rxy)
+    f = C.polynomial
     r = f.degree()
-    m = curve.exponent
+    m = C.exponent
     g = Fxy(g)
     g1 = g.numerator()
     g2 = g.denominator()
     
-    Rx.<x> = PolynomialRing(F)
+    Rx.<x1> = PolynomialRing(F)
     Fx = FractionField(Rx)
-    FxRy1.<y> = PolynomialRing(Fx, 1) #coercion problems from Rxy to FxRy
-    FxRy.<y2> = PolynomialRing(Fx) #coercion problems from Rxy to FxRy
-    (A, B, C) = xgcd(FxRy(FxRy1(g2)(y = y2)), FxRy(FxRy1(y^m - f)(y = y2))) #coercion problems from Rxy to FxRy
-    g = FxRy(FxRy1(g1)(y = y2))*B/A
-    g = FxRy(g)
+    FxRy.<y1> = PolynomialRing(Fx)    
+    (A, B, C) = xgcd(FxRy(g2(x=x1, y=y1)), y1^m - FxRy(f(x=x1)))
+    g = FxRy(FxRy(g1(x=x1, y=y1))*B/A)
     
-    while(g.degree(y2) >= m):
-        d = g.degree(y2)
+    while(g.degree(y1) >= m):
+        d = g.degree(y1)
         G = coff(g, d)
         i = floor(d/m)
-        g = g - G*y2^d + f^i * y2^(d%m) *G
-    Rxy1.<x3, y3> = PolynomialRing(F, 2)
-    Fxy1 = FractionField(Rxy1)
-    g = sum(Fxy(y3)^i*Fx(coff(g, i)) for i in range(0, m))
-    g = Fxy(g)
-    return(g)
+        g = g - G*y1^d + Rx(f(x=x1)^i) * y1^(d%m) *G
+    
+    return(Fxy(g(x1=x, y1 = y)))
 
 def reduction_form(C, g):
     '''Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)
@@ -284,18 +280,17 @@ def reduction_form(C, g):
     m = C.exponent
     g = reduction(C, g)
 
-    g1 = Rxy(0)
-    Rx.<x> = PolynomialRing(F)
+    Rx.<x1> = PolynomialRing(F)
     Fx = FractionField(Rx)
-    FxRy1.<y> = PolynomialRing(Fx, 1)
-    FxRy.<y2> = PolynomialRing(Fx)
-    
-    g = FxRy(FxRy1(g)(y = y2))
+    FxRy.<y1> = PolynomialRing(Fx)
+    g1 = FxRy(0)
+    g = FxRy(g(x = x1, y=y1))
     for j in range(0, m):
         if j==0:
             G = coff(g, 0)
-            g1 += Fxy(Fx(G))
+            g1 += FxRy(G(x=x1))
         else:
             G = coff(g, j)
-            g1 += Fxy(y)^(j-m)*Fxy(Fx(f*G))
+            g1 += y1^(j-m)*FxRy(f(x=x1)*G)
+    g1 = Fxy(g1(x1 = x, y1 = y))
     return(g1)
\ No newline at end of file