Lint fixes and more

Author: whoami730

src/sage/modules/free_module_integer.py

@@ -901,11 +901,11 @@ def hadamard_ratio(self, use_reduced_basis=True):
         r"""
         Computes the normalized Hadamard ratio of the given basis.
 
-        The normalized Hadamard ratio of the basis `B = {v_1, v_2, \dots, v_n}` is defined as
+        The normalized Hadamard ratio of the basis `B = \left{ v_1, v_2, \dots, v_n \right}` is defined as
 
         .. MATH::
 
-            \mathcal{H}(B) = \dfrac{det L}{\|v_1\| \|v_2\| \cdots \|v_n\|}^{\dfrac{1}{n}}
+            \mathcal{H}(B) = \left( \dfrac{\det L}{\|v_1\| \|v_2\| \cdots \|v_n\|} \right)^{\frac{1}{n}}
 
         The closer this ratio is to 1, the more orthogonal the basis is.
 
@@ -932,23 +932,24 @@ def hadamard_ratio(self, use_reduced_basis=True):
             basis = self.reduced_basis
         else:
             basis = self.basis_matrix()
-        
+
         n = basis.nrows()
         r = self.rank()
         assert r == n
-        
+
         ratio = (self.discriminant().sqrt() / prod([v.norm() for v in basis]))**(1/r)
         assert 0 < ratio <= 1
         return ratio
-    
+
     def gaussian_heuristic(self, exact_form=False):
         r"""
-        Computes the Gaussian expected shortest length, also known as the Gaussian  
+        Computes the Gaussian expected shortest length, also known as the Gaussian
         heuristic. This estimates the expected norm of the shortest non-zero vector
         in the lattice. The heuristic is independent of the chosen basis.
 
         INPUT:
-        - ``exact_form`` -- boolean (default: ``False``); uses exact formulation based on gamma function, instead of estimation of the gamma function
+        - ``exact_form`` -- boolean (default: ``False``); uses exact formulation
+          based on gamma function, instead of using stirling's approximation
 
         OUTPUT: The Gaussian heuristic described above.
 
@@ -965,13 +966,13 @@ def gaussian_heuristic(self, exact_form=False):
             21.375859827168494
         """
         basis = self.basis_matrix()
-        
+
         n = basis.nrows()
         r = self.rank()
         assert r == n
 
         D = self.discriminant().sqrt()
-        
+
         if exact_form:
             return (D * gamma(1 + (r/2)))**(1/r) / pi.sqrt()
         else: