We give an algorithm to compute Cl(L) and we describe its initial ideal under the order filtration. Our main application is an algorithm for constructing a Jordan-Holder series for a holonomic D-module and a formula for its length. Using the closure, we also reproduce a result of Strombeck (1978), who described the initial ideals of left ideals of D under the order filtration, and a result of Cannings and Holland (1994), who described the isomorphism classes of right ideals of D.
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