Does the rank of an elliptic curve remain same under any group law?
Say, we have an elliptic curve of rank 2 and we are working over rationals and we are using lines to produce other rational points.
Then if I use the same elliptic curve but instead of using lines I use some other curves to produce other points (using different group law), is the number of independent points again 2?
EDIT: Answer to dan_fulea's comment:
If I use parabolas instead of lines:
$$E : y^2=x^3+17^2x$$ $$C_1 : y=\frac{863 x^2}{18496}+\frac{803 x}{136}-\frac{157}{4}$$ $$C_2 : y=\frac{231 x^2}{5776}+\frac{9835 x}{1444}-\frac{25296}{361}$$ $$C_3 : y=\frac{12 x^2}{247}+\frac{2465 x}{494}+\frac{3468}{247}$$ Known points $[P_1, P_2, P_3]$: $$P_1=(68,578),P_2=(144,1740),P_3=(\frac{17}{4},\frac{289}{8})$$ New points constructed by intersections of $E$ with $C_1$, $C_2$, $C_3$, $[N_1, N_2, N_3]$: $$N_1=(\frac{1676132}{744769},-\frac{16534737914}{642735647}),N_2=(\frac{246016}{53361},-\frac{466192880}{12326391}),N_3=(\frac{289}{144},\frac{41905}{1728})$$ $$C_1 \cap E=[P_1, P_1, P_1, N_1]$$ $$C_2 \cap E=[P_1, P_1, P_2, N_2]$$ $$C_3 \cap E=[P_1, P_2, P_3, N_3]$$
I could have used other curves not just parabolas with even more contacts.
But I now see that all this can be done using lines. For example $N_1=-3*P_1$. So perhaps I need as many independent points with parabolas as I would need with lines.