This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.

What motivated the use of the word "conductor" in the first place?

A friend of mine once pointed out the amusing idea that one can think of the conductor of an elliptic curves as "someone" driving a train which lets you off at the level of the associated modular form.

A similar statement can be made concerning Szpiro's conjecture, which provides asymptotic bounds on several invariants of an elliptic curve in terms of its conductor. Here one might think of the conductor as "someone" who controls this symphony of invariants consisting of the minimal discriminant, the real period, the modular degree, and the order of the Shafarevich-Tate group (assuming BSD).

Was there some statement of this sort which motivated Artin's original definition of the conductor?

Does anyone have a reference for the first appearance of the word conductor in this context?

I apologize if this question is inappropriate for MO.