Journal of the
Korean Mathematical Society JKMS

A simple proof of the fact that the $j$-invariants for Weierstrass equations are invariant under birational transformations which keep the forms of Weierstrass equations is given by finding a non-trivial explicit birational transformation which sends a normalized Weierstrass equation to the same equation.

Keywords: $j$-invariant, Weierstrass equation, elliptic curve, coverings, Riemann surface of genus one

MSC numbers: Primary 14H52; Secondly 30F10, 14H55