Отправить post запрос на сайт, так, чтобы авторизоваться и сохранить куки

HttpWebRequest httpWebRequest = HttpWebRequest.Create("http://www.site.ru/login") as HttpWebRequest;
httpWebRequest.Method = "POST";
httpWebRequest.ContentType = "application/x-www-form-urlencoded";
httpWebRequest.UserAgent = "Mozilla/5.0 (Windows NT 6.1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/38.0.2125.122 YaBrowser/14.12.2125.10034 Safari/537.36";
HttpWebResponse httpWebResponse = httpWebRequest.GetResponse() as HttpWebResponse;
           
CookieContainer cookies = new CookieContainer();
httpWebRequest.CookieContainer = cookies;
 
using (StreamWriter sw = new StreamWriter(httpWebRequest.GetRequestStream()))
{
    sw.Write("auth_key=880ea6a14ea49e853634fbdc5015a024&email=email&pass=123");
}
            
using (StreamReader sr = new StreamReader(httpWebResponse.GetResponseStream()))
{
 
     MessageBox.Show(sr.ReadToEnd());
 
}
 
cookies = httpWebRequest.CookieContainer;