Szpiro's conjecture
In number theory, Szpiro's conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve.  In a general form, it is equivalent to the well-known abc conjecture.  It is named for Lucien Szpiro who formulated it in the 1980s.

Statement
The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

Modified Szpiro conjecture
The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f (using notation from Tate's algorithm), we have