34212: log embedding for number field elements

src/sage/rings/number_field/number_field_element.pyx

@@ -4117,6 +4117,34 @@ cdef class NumberFieldElement(FieldElement):
         """
         return (self.global_height_non_arch(prec)+self.global_height_arch(prec))/self.number_field().absolute_degree()
 
+    def log_embedding(self, x, prec=53):
+        """
+        Return the logarithmic embedding of ``x``.
+
+        The logarithmic embedding is a tuple of real embeddings and conjugate
+        pairs of compelx embeddings.
+        It is defined under Definition 4.9.6 in [Cohen1993]_.
+        """
+        K = self.base_ring()
+        K_embeddings = K.places(prec)
+        r1, r2 = K.signature()
+        r = r1 + r2 - 1
+
+        Reals = RealField(prec)
+
+        if x == 0:
+            return vector([-1 for _ in range(r + 1)])
+
+        x_logs = []
+        for i in range(r1):
+            sigma = K_embeddings[i]
+            x_logs.append(Reals(abs(sigma(x))).log())
+        for i in range(r1, r + 1):
+            tau = K_embeddings[i]
+            x_logs.append(2 * Reals(abs(tau(x))).log())
+
+        return vector(x_logs)
+
     def numerator_ideal(self):
         """
         Return the numerator ideal of this number field element.