Each LED has a vDrop of 2.4v and they are wired in series with a 40 ohm 5w resistor.
Did you measure this voltage from a single LED with 40 ohm resistor on the 12V battery? If each LED drops 2.4V than 20 of them in series with do 48V. Tell more. The answer to your question might simply be a MOSFET, no need for a relay.
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