Error:-

1.closing });missed of $("#email").change(function(){

Working example code:-

$("#email").change(function(){
  alert('email changed');
  var Email = $('#email').val();
  var formsubmission = 'http://127.0.0.1/heropet/Common/check_mail';
  $.ajax({
    method: "POST",
    url: formsubmission,
    data: {email:Email},
    success: function(response){
      alert(response);
    }
  });
 });

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<label>E-mail</label>
<input type="text" name="email" id="email" value=""><br>

Note:- add a proper jquery library as i have added into my example.

You are missing "})" at the end.

Also wrap it in $( document ).ready(function(){})

As shown below:

    $( document ).ready(function(){
    $("#email").change(function(){
        alert('email changed');
        var Email = {
            email: $('#email').val()
        }
        var formsubmission = 'http://127.0.0.1/heropet/Common/check_mail';
        $.ajax({
            method: "POST",
            url: formsubmission,
            data: Email,
            success: function(response){
                alert(response);
            }
        });
    });
});