Enter in Textbox reloads page instead of submitting form

[Toot]

I'm sure this is simple and there's plenty of close-but-not-quite posts...

I've got a content page containing a LoginView which has got a Textbox and Button on the AnonymousTemplate. I enter "123" in the Textbox and press enter, the page reloads (I've put a trace in the Page_Load event) and the Textbox still contains "123". If I press my Button instead of pressing enter while in the Textbox, it works as I would expect. I've tried adding

ClientScript.RegisterHiddenField("__EVENTTARGET", "MyButtonName");

(suggested instead of Page.RegisterHiddenField by my compiler) but that made no difference.
What's going on? How do I get the expected behaviour instead of a page reload?

Thanks for your help!