No one else know or understand your system setup better than yourself.  The issue was just happened out of the blue or following an update, upgrade.  Did you give it a new coin cell batt while clearing cmos.  Go over the resolution for error codes 123 again to see anything amiss.

We tried reaching you on a private message asking for the Service Tag number to ascertain the warranty but did not receive a response. Please feel free to reply to the private message whenever you are available.

Hello Chino de Oro,

I changed some bios settings, like boot order and autostart on power recovery.
I cannot process the last two points since I have no display.

That's why I'm trying to clear CMOS.

Any symptoms with the machine operations or any type of works being done on the machine prior to the error codes and no display issue?  Those info could help with troubleshooting and direction to finding solution. 

If no display, how did you make changes in BIOS?  Or you meant that you made change in BIOS and it caused the issue of no display? 

The display was functional and I had no particular problems.
I had to boot from a usb key, that's why I modified the bios.
After this modification, I lost the display and the error code appeared.

While pulling the battery, replace it with a fresh new coin cell.  You can try the rtc reset jumper method as it will work at power rails.

Hello,
I have already replaced the battery and done the CMOS reset with the RTC jumper, without result.
I think the reset worked because my T5610 don't start when the power returns, but I still have no display.

Try to test with a different monitor.  

The monitor is ok, it is running with another PC, each input  is ok (VGA, DVI & HDMI).

May start with all components removal test.  Review these articles for references and testing

No POST troubleshooting

Computer does not start

Reference guide to workstation diagnostic indicators