We know that the constructible universe $L$ is an absolute and minimal model of ZF (every standard model of ZF contains "an" $L$, and it is actually the same $L$ for all of them).
It is also my understanding that the existence of $0\sharp$ informally means that $V$ is much "bigger" than $L$ (meaning that if $0\sharp$ exists then even $\aleph_1$ is already an inaccessible cardinal in $L$) and that a sort of converse is true (see: https://en.wikipedia.org/wiki/Jensen%27s_covering_theorem).
Therefore my question is: Is there an absolute minimal model of ZF + $\exists0\sharp$ ? A sort of "$L\sharp$" if we really want to abuse notation?