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A heavy-fermion superconductor UPt$_3$ is a unique spin-tripletsuperconductor with multiple superconducting phases. Here we provide the firstreport on the first-principles analysis of the microscopic superconducting gapstructure. We find that the promising gap structure is an unprecedented$E_{2u}$ state, which is completely different from the previousphenomenological $E_{2u}$ models. Our obtained $E_{2u}$ state has in-planetwofold vertical line nodes on small Fermi surfaces and point nodes with lineardispersion on a large Fermi surface. These peculiar features cannot beexplained in the conventional spin $1/2$ representation, but is described bythe group-theoretical representation of the Cooper pairs in the total angularmomentum $j=5/2$ space. Our findings shed new light on the long-standingproblems in the superconductivity of UPt$_3$

Exotic multi-gap structure in UPt$_3$ unveiled by the first-priniciples analysis