Add test if quaternion order is maximal

src/sage/algebras/quatalg/quaternion_algebra.py

@@ -77,6 +77,7 @@
 from sage.misc.cachefunc import cached_method
 
 from sage.categories.algebras import Algebras
+from sage.categories.number_fields import NumberFields
 
 ########################################################
 # Constructor
@@ -1758,6 +1759,42 @@ def discriminant(self):
 
         return (MatrixSpace(QQ, 4, 4)(L)).determinant().sqrt()
 
+    def is_maximal(self):
+        r"""
+        Check whether the order of ``self`` is maximal in the ambient quaternion algebra.
+
+        Only works in quaternion algebras over number fields
+
+        OUTPUT: Boolean
+
+        EXAMPLES::
+
+            sage: p = 11
+            sage: B = QuaternionAlgebra(QQ, -1, -p)
+            sage: i, j, k = B.gens()
+            sage: O0_basis = (1, i, (i+j)/2, (1+i*j)/2)
+            sage: O0 = B.quaternion_order(O0_basis)
+            sage: O0.is_maximal()
+            True
+            sage: O1 = B.quaternion_order([1, i, j, i*j])
+            sage: O1.is_maximal()
+            False
+
+        TESTS::
+
+            sage: B = QuaternionAlgebra(GF(13), -1, -11)
+            sage: i, j, k = B.gens()
+            sage: O0_basis = (1, i, j, k)
+            sage: O0 = B.quaternion_order(O0_basis)
+            sage: O0.is_maximal()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: check for maximality is only implemented for quaternion algebras over number fields
+        """
+        if self.quaternion_algebra().base_ring() not in NumberFields():
+            raise NotImplementedError("check for maximality is only implemented for quaternion algebras over number fields")
+        return self.discriminant() == self.quaternion_algebra().discriminant()
+
     def left_ideal(self, gens, check=True, *, is_basis=False):
         r"""
         Return the left ideal of this order generated by the given generators.