If $ f = a_{n}x^{n} + a_{n-1}x^{n-1}+ \dots + a_{0} = a_{n} \prod^{n}_{i=1}(x-\alpha_{i}), $ where $ a_{n} \neq 0, $ then the discriminant of $ f $ is defined as $$ \text{Disc}(f) := a_{n}^{2n-2} \prod_{i \neq j}(\alpha_{i}-\alpha_{j}). $$
The Weierstass form of some cubic is $ y^2=4x^3 -g_{2}x-g_{3}. $ Consider the corresponding projective curve $$ C_{g_{2},g_{3}} : x_{0}x^{2}_{2}-4x^{3}_{1}+g_{2}x_{1}x^{2}_{0}+g_{3}x^{3}_{0}. $$
Klaus Hulek says on page 129 of his "Elementary Algebraic Geometry" that: "we define the discriminant of $ C_{g_{2},g_{3}} $ to be the discriminant of $ 4x^{3}+g_{2}x+g_{3} $ divided by 16."
I realise that this is a definition, but I wonder what possible reason there may be for this, as it is a rather specific definition.