You considered an index fossil it has to have lived for a relatively short period of time

Yes, an index fossil is a type of fossil that is used to define and identify a particular time period in geologic history. It must fulfill certain criteria, one of which is having lived for a relatively short period of time. This means that the fossil organism must have been abundant, widespread, and existed for a specific interval of time that can be easily recognized and correlated with other rock layers or fossils. The short duration of the fossil's existence ensures that it can be used as a reliable indicator of a specific time period.