diff --git a/src/sage/rings/padics/witt_vector.py b/src/sage/rings/padics/witt_vector.py
index 4712732b9da..711d1c6686c 100644
--- a/src/sage/rings/padics/witt_vector.py
+++ b/src/sage/rings/padics/witt_vector.py
@@ -83,10 +83,9 @@ def __init__(self, parent, vec=None):
         self._prec = parent.precision()
         B = parent.coefficient_ring()
         if vec is not None:
-            if isinstance(vec, int) or isinstance(vec, Integer):
+            if isinstance(vec, (int, Integer)):
                 self._int_to_vector(vec, parent)
-            elif (isinstance(vec, tuple) or isinstance(vec, list)
-                    or isinstance(vec, WittVector)):
+            elif isinstance(vec, (tuple, list, WittVector)):
                 if len(vec) < self._prec:
                     raise ValueError(f"{vec} has not the correct length, "
                                      "expected length has to be at least "
@@ -420,15 +419,13 @@ def __init__(self, parent, vec=None, phantom=None):
         R = parent.coefficient_ring()
         p = parent.prime()
         base = R
-        if (isinstance(R, PolynomialRing_generic)
-                or isinstance(R, MPolynomialRing_base)):
+        if isinstance(R, (PolynomialRing_generic, MPolynomialRing_base)):
             base = R.base()
         base_lift = QqFP(base.cardinality(), prec=self._prec,
                          modulus=base.modulus(), names=(base.variable_name(),),
                          res_name=base.variable_name())
         lift = base_lift
-        if (isinstance(R, PolynomialRing_generic)
-                or isinstance(R, MPolynomialRing_base)):
+        if isinstance(R, (PolynomialRing_generic, MPolynomialRing_base)):
             lift = R.change_ring(base_lift)
         if phantom is not None:
             self._phantom = phantom
@@ -442,13 +439,12 @@ def __init__(self, parent, vec=None, phantom=None):
             self._coordinates = vec.coordinates()
             self._phantom = vec._phantom
             self._powers = vec._powers
-        elif isinstance(vec, int) or isinstance(vec, Integer):
+        elif isinstance(vec, (int, Integer)):
             self._int_to_vector(vec, parent)
             y = base_lift(vec)
             self._powers = [y]
             self._phantom = self._prec * [y]
-        elif (isinstance(vec, tuple) or isinstance(vec, list)
-                or isinstance(vec, WittVector)):
+        elif isinstance(vec, (tuple, list, WittVector)):
             if len(vec) < self._prec:
                 raise ValueError(f"{vec} has not the correct length, "
                                  "expected length has to be at least "
diff --git a/src/sage/rings/padics/witt_vector_ring.py b/src/sage/rings/padics/witt_vector_ring.py
index c4630f51b44..9b86c92d962 100644
--- a/src/sage/rings/padics/witt_vector_ring.py
+++ b/src/sage/rings/padics/witt_vector_ring.py
@@ -217,7 +217,7 @@ def __classcall_private__(cls, coefficient_ring, prec=1, p=None, algorithm=None)
         """
         if coefficient_ring not in CommutativeRings():
             raise TypeError(f"{coefficient_ring} is not a commutative ring")
-        elif not (isinstance(prec, int) or isinstance(prec, Integer)):
+        elif not isinstance(prec, (int, Integer)):
             raise TypeError(f"{prec} is not an integer")
         elif prec <= 0:
             raise ValueError(f"{prec} must be positive")
@@ -801,7 +801,7 @@ class WittVectorRing_phantom(WittVectorRing):
         sage: W = WittVectorRing(QQ, p=23, prec=3, algorithm='phantom')
         Traceback (most recent call last):
         ...
-        ValueError: the 'phantom' algorithm only works when the coefficient ring is a finite field of p, or a polynomial ring on that field
+        ValueError: the 'phantom' algorithm only works when the coefficient ring is a finite field of char. p, or a polynomial ring on that field
     """
     Element = WittVector_phantom
 
@@ -819,15 +819,17 @@ def __init__(self, coefficient_ring, prec, prime) -> None:
 
             sage: TestSuite(W).run()
         """
-        if not (coefficient_ring.characteristic() == prime
-                and (coefficient_ring in Fields().Finite()
-                     or ((isinstance(coefficient_ring, PolynomialRing_generic)
-                          or isinstance(coefficient_ring,
-                                        MPolynomialRing_base))
-                         and coefficient_ring.base() in Fields().Finite()))):
-            raise ValueError("the 'phantom' algorithm only works when the "
-                             "coefficient ring is a finite field of "
-                             "p, or a polynomial ring on that field")
+        msg = "the 'phantom' algorithm only works when the coefficient ring is"\
+            " a finite field of char. p, or a polynomial ring on that field"
+
+        if coefficient_ring.characteristic() != prime:
+            raise ValueError(msg)
+
+        if not (coefficient_ring in Fields().Finite() or
+                (isinstance(coefficient_ring, (PolynomialRing_generic,
+                                               MPolynomialRing_base)) and
+                 coefficient_ring.base() in Fields().Finite())):
+            raise ValueError(msg)
 
         if (coefficient_ring in Fields().Finite()
             or isinstance(coefficient_ring,