Is it permissible to pray Fajr at home – before the prayer is established in the mosque – when feeling extremely tired due to staying up late studying?

The fundamental principle is that a Muslim is obligated to pray in the mosque with the congregation. Whoever hears the call to prayer and does not attend, has no prayer except if there is an excuse. If one has an excuse, then there is no blame upon them for praying with their household in congregation. If there is no one to pray with, then they should pray alone after the time for prayer has begun. Exhaustion is not an excuse for abandoning the congregation unless it reaches an extreme level. However, one should not exhaust themselves by staying up late and then miss the congregation, for the Messenger of Allah, peace and blessings be upon him, said: "The prayer of one of you in congregation is twenty-odd degrees more than his prayer in his marketplace and his house."