If this is just about presentation, you have a couple of options:
1) Use attribute notation instead of association edges. From the UML 2.4.1 Superstructure spec:
Figure 7.24 shows that the attribute notation can be used for an association end owned by a class, because an association end owned by a class is also an attribute.
2) Just break it down into many small diagrams with just 3-4 classes each.
3) Compartmentalize your diagram. Many UML tools allow a class to appear in a diagram more than once. Even if you don't want many small diagrams (eg. because you want to have all the classes appear in a single diagram), you can break it down into smaller groups and thus avoid a big tangly mess of edges.