Laravel 5 relationship belongsTo not working - php

Question

First of all what I want to achieve it's quite simple : I want to use relationship belongsTo from Laravel 5 to get a value from another table. Like so
<?= $email->contact->first_name; ?>
But this is not working at all, instead, I get this error : Trying to get property of non-object. But when I do print_r($email) it shows me that $email is actually an object...So I really don't understand now.
This is (a part of) what I have in my Controller :
$emails = Email::where('user_id', '=', Auth::user()->id)->take(6)->orderBy('created_at', 'desc')->get();
This is what I have in my view :
foreach($emails as $key => $email) {
echo $email->contact->first_name; ?>
}
This is my model email
public function contact()
{
return $this->belongsTo('App\Contact', 'contact_id');
}
As you can see, I am using a second parameter inside belongsTo method, which is 'contact_id' because I don't want to use 'user_id' as the foreign key.
EDIT:
This is my output of print_r($email) :
App\Email Object ( [connection:protected] => [table:protected] => [primaryKey:protected] => id [perPage:protected] => 15 [incrementing] => 1 [timestamps] => 1 [attributes:protected] => Array ( [id] => 88 [user_id] => 1 [group_id] => 1 [contact_id] => 79 [reminder_id] => 31 [subject] => Test task #3 [view] => 0 [view_hash] => $2y$10$NZ3B.UUqtRRl2opH5UwXOeRNeRROeZGdid6Crs4ShT1VptbZJS.iu [content] => blablabla
EDIT 2 :
This is my output of print_r($email->contact) :
App\Contact Object ( [fillable:protected] => Array ( [0] => title [1] => content ) [table:protected] => contacts [connection:protected] => [primaryKey:protected] => id [perPage:protected] => 15 [incrementing] => 1 [timestamps] => 1 [attributes:protected] => Array ( [id] => 86 [user_id] => 1 [group_id] => 1 [first_name] => Matthieu [last_name] => Boistoule [email] => m.boistoule#gmail.com [phone] => [company] => [title] => [website] => [industry] => [address] => [city] => [country] => [province] => [postal_code] => [lead_source] => [notes] => [created_at] => 2015-06-04 12:13:00 [updated_at] => 2015-06-04 12:13:00 ) [original:protected] => Array ( [id] => 86 [user_id] => 1 [group_id] => 1 [first_name] => Matthieu [last_name] => Boistoule [email] => m.boistoule#gmail.com [phone] => [company] => [title] => [website] => [industry] => [address] => [city] => [country] => [province] => [postal_code] => [lead_source] => [notes] => [created_at] => 2015-06-04 12:13:00 [updated_at] => 2015-06-04 12:13:00 ) [relations:protected] => Array ( ) [hidden:protected] => Array ( ) [visible:protected] => Array ( ) [appends:protected] => Array ( ) [guarded:protected] => Array ( [0] => * ) [dates:protected] => Array ( ) [casts:protected] => Array ( ) [touches:protected] => Array ( ) [observables:protected] => Array ( ) [with:protected] => Array ( ) [morphClass:protected] => [exists] => 1 )