chmod 777 no effects on Linux Mint

NTFS does not handle file permissions in the same was as Linux, as explained in the ntfs-3g FAQ:

Why have chmod and chown no effect? [sic]

By default files on NTFS are owned by root with full access to everyone. To get standard per-file protection you should mount with the “permissions” option. Moreover, if you want the permissions to be interoperable with a specific Windows configuration, you have to map the users.

So, either modify your /etc/fstab with the appropriate options: (I changed default_permissions to permissions):

/dev/sda4 /media/DATA ntfs rw,nosuid,nodev,allow_other,permissions,blksize=4096

or, just unmount and remount manually:

sudo umount /media/DATA
sudo mount -t ntfs -o rw,nosuid,nodev,relatime,permissions /dev/sda4 /media/DATA

You should now be able to change permissions normally with sudo chmod.