How can we decide two elliptic curves over Q are isomorphic over Q?

Question

How can we decide which of the following elliptic curves over $\mathbb{Q}$ are isomorphic over $\mathbb{Q}$? $$E_1:y^2=x^3+1$$ $$E_2:y^2=x^3+2$$ $$E_3:y^2=x^3+x+1$$

Answer 1

See Silverman's book and answer there namely given $$E_i/K:y^2=x^3+a_ix+b_i,\qquad a_ib_i\ne 0$$ for $char(K)\ne 2,3$

Since $E_i\cong y^2=x^3+c^4 a_ix+c^6 b_i$ and $j(E_i) = 1728 \frac{-4 a_i^3}{-4a_i^3-27b_i^2}=1728 \frac{-4}{-4-27(b_i/a_i)^2 /a_i}$

It suffices to compare $j(E_i)\in K$ and $a_i/b_i\in K^*/K^{*2}$ to find if $E_i\cong E_l$ over $K$

($K^*/K^{*4},K^*/K^{*6}$ in the few degenerate cases $a_ib_i=0$ ie. $j\in 0,1728$)