  Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a
number field with either good ordinary reduction or multiplicative reduction at
each prime of $F$ above $p$. We shall study the characteristic element of the
Selmer group of $E$ over a $p$-adic Lie extension. In particular, we relate the
order of vanishing of these characteristic element evaluated at Artin
representations to the Selmer coranks and their twists in the intermediate
subextensions of the $p$-adic Lie extension.
