There is a .ASPXAUTH cookie set, you are obviously correct. It is used to determine if a user if logged in.
To get what you need look over your web.config for the config section:
<authentication mode="Forms">
<forms
loginUrl="~/login.aspx"
protection="All"
timeout="30"
name="ExampleSite.FormsAuthentication"
path="/"
requireSSL="false"
slidingExpiration="true"
defaultUrl="index.aspx"
cookieless="UseDeviceProfile"
enableCrossAppRedirects="false"
/>
</authentication>
When the user is successfully authenticated a cookie will be set based off the name="ExampleSite.FormsAuthentication" parameter. It will expire after logging out or after the session expires. You will see a cookie on Chrome/FFX or whatever browser you are using called ExampleSite.FormsAuthentication with an encrypted value. Obviously the name parameter you are using will be different and not ExampleSite.FormsAuthentication but you get the idea.
You could always check and see if the cookie exists. As mentioned be careful of the http-only (with relation to JS). As you can also override that value in the web.config so you can access it with JS.
<httpCookies httpOnlyCookies="false" requireSSL="false" domain="" />