Getting only the login field
In your theme's template.php file:

/**
 * Implements hook_form_FORM_ID_alter().
 */

function YOURTHEMENAME_form_user_login_block_alter(&amp;$form) {

    unset($form['links']);
    return;

}


In this function you have access to the form as an array. Usually I enable devel module and do a dsm($form) . In your case dsm() will not work because you need to be logged in to be able to see the output. I used var_dump($form) to see the form and used unset() function to remove the links.