gh-37157: Add additional lattice methods for matrices over QQ
    
LLL has long been implemented for matrices over QQ by clearing the
denominator and doing LLL over ZZ, but this was not done for BKZ() or
is_LLL_reduced()

This PR adds BKZ() and is_LLL_reduced() to rational matrices, using the
same techniques as the current LLL method.
    
URL: #37157
Reported by: TheBlupper
Reviewer(s): grhkm21

Author: Release Manager

src/sage/matrix/matrix_integer_dense.pyx

@@ -2746,7 +2746,8 @@ cdef class Matrix_integer_dense(Matrix_dense):
     def BKZ(self, delta=None, algorithm="fpLLL", fp=None, block_size=10, prune=0,
             use_givens=False, precision=0, proof=None, **kwds):
         """
-        Block Korkin-Zolotarev reduction.
+        Return the result of running Block Korkin-Zolotarev reduction on
+        ``self`` interpreted as a lattice.
 
         INPUT:
 

src/sage/matrix/matrix_rational_dense.pyx

@@ -2927,13 +2927,39 @@ cdef class Matrix_rational_dense(Matrix_dense):
     # LLL
     # ###############################################
 
+    def BKZ(self, *args, **kwargs):
+        """
+        Return the result of running Block Korkin-Zolotarev reduction on
+        ``self`` interpreted as a lattice.
+
+        The arguments ``*args`` and ``**kwargs`` are passed onto
+        :meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.BKZ`,
+        see there for more details.
+
+        EXAMPLES::
+
+            sage: A = Matrix(QQ, 3, 3, [1/n for n in range(1, 10)])
+            sage: A.BKZ()
+            [ 1/28 -1/40 -1/18]
+            [ 1/28 -1/40  1/18]
+            [-1/14 -1/40     0]
+
+            sage: A = random_matrix(QQ, 10, 10)
+            sage: d = lcm(a.denom() for a in A.list())
+            sage: A.BKZ() == (A * d).change_ring(ZZ).BKZ() / d
+            True
+        """
+        A, d = self._clear_denom()
+        return A.BKZ(*args, **kwargs) / d
+
     def LLL(self, *args, **kwargs):
         """
         Return an LLL reduced or approximated LLL reduced lattice for
         ``self`` interpreted as a lattice.
 
-        For details on input parameters, see
-        :meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.LLL`.
+        The arguments ``*args`` and ``**kwargs`` are passed onto
+        :meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.LLL`,
+        see there for more details.
 
         EXAMPLES::
 
@@ -2942,10 +2968,33 @@ cdef class Matrix_rational_dense(Matrix_dense):
             [ 1/28 -1/40 -1/18]
             [ 1/28 -1/40  1/18]
             [    0 -3/40     0]
+
+            sage: A = random_matrix(QQ, 10, 10)
+            sage: d = lcm(a.denom() for a in A.list())
+            sage: A.LLL() == (A * d).change_ring(ZZ).LLL() / d
+            True
         """
         A, d = self._clear_denom()
         return A.LLL(*args, **kwargs) / d
 
+    def is_LLL_reduced(self, delta=None, eta=None):
+        r"""
+        Return ``True`` if this lattice is `(\delta, \eta)`-LLL reduced.
+        For a definition of LLL reduction, see
+        :meth:`sage.matrix.matrix_integer_dense.Matrix_integer_dense.LLL`.
+
+        EXAMPLES::
+
+            sage: A = random_matrix(QQ, 10, 10)
+            sage: L = A.LLL()
+            sage: A.is_LLL_reduced()
+            False
+            sage: L.is_LLL_reduced()
+            True
+        """
+        A, _ = self._clear_denom()
+        return A.is_LLL_reduced(delta, eta)
+
 
 cdef new_matrix_from_pari_GEN(parent, GEN d) noexcept:
     """