Both your Panasonic KX-TGA510M phone and its corresponding base have independent ringer functions, and you can turn the ringer on or off for either. For bases with multiple registered handsets, you must adjust the ringer on each handset individually, though you cannot register more than a total of eight handsets to a single base.
Press the “CONF/FUNCTION” button on the bottom-left of the handset's keypad, just below the Tone button. Press the up and down arrow keys until you reach the “Ringer setting” option, and select that by pressing the right arrow key. To turn off the ringer, press the down arrow key until you see “Off?” on the screen, and then press the soft key for “SAVE,” which is just below the display. You can turn the ringer on using the same method, pressing the up arrow instead of the down one for the Ringer setting.
Press the “FUNCTION/EDIT” key on the base unit; it is above and to the right of the arrow navigation keys. Press the up and down arrow keys until “Ringer volume” appears on the display, and then press the right navigation key. Press the down arrow repeatedly until “Off?” appears on the display, and then press the right navigation key to save this. To turn the ringer on, use the up navigation key at the Ringer setting screen rather than the down navigation key.