Consider solution of $$x^3-y^2=1728 \text{ unit} \qquad (1)$$ in a number field.

This is related to the discriminant of elliptic curve in terms of $c_4,c_6$.

Via elliptic curves it might have infinitely solutions for fixed unit.

I am interested if $x,y$ are in the ring of integers of the number field.

If one solution exists, scaling by powers of units gives infinitely, so define a set of solutions to be primitive if the norms of $x$ are distinct.

Q1 Does (1) have infinitely many primitively solutions when $x,y$ are in the ring of integers?

Q2 Does positive solution to Q1 contradicts the abc conjecture over number fields?

Q3 What is the expected abc quality of triples resulting from (1)? Say for $w=\sqrt{2}$,

`A= (w + 1) * 3^3 ;B= (-70*w - 99) * w; C= (-2*w - 1)^4, A+B+C=0`

I suppose this is impossible since it might solve $x^3-y^2=u^n$ for many $n$.