A minimum energy transfer orbit is a hohmann transfer orbit. A hohmann transfer orbit requires the smallest amount of delta V as the first burn is at periapsis (closes point to body being orbited) and the second burn is at apoapsis. For a planetary transfer the minimum energy launch windows occur at intervals of the target planets synodic period in relation to the departure planet.

No one can answer this question until we know more details about how the trip is to be conducted. For example, when NASA launched men to the Moon, they used something called a Hohmann transfer orbit. But recently, the Indian government sent a probe to the Moon by a means that used much less energy than needed for a Hohmann transfer orbit, but which took much more time. So there is no one answer to this question.

145.9 days

The earth needs to be at a point in its orbit from which the Hohmann transfer orbit between earth and the orbit of Saturn
requiring the least launch energy terminates at the point where Saturn will be when the vehicle gets there.

You might want to read that again, slowly. That's as plain as I know how to say it.

Martin Hohmann was born in 1948.