It is in general very difficult to compute ranks of elliptic curves over number fields, even if equipped with any conjectures that are available. On the other hand, the parity of the rank is (conjecturally) very easy to determine — it is given as a sum of purely local terms, which have a reasonably simple classification. Since "odd rank" implies "non-zero rank" implies "the curve has infinitely many points", this leads to a number of (conjectural!) arithmetic phenomena.