Let $Y_1(n)$ (for $n\ge 4$) be the fine moduli scheme over $\mathbb{Q}$ parametrizing elliptic curves with a rational point of order $n$. Let $\mathbb{A}^1_j$ be the $j$-line over $\mathbb{Q}$, the coarse moduli scheme of the moduli stack of elliptic curves $\mathcal{M}_{1,1}$ (over $\mathbb{Q}$). Let $\mathcal{M}_{1,1}^\circ$ be the open substack classifying elliptic curves with $j$-invariant $\notin\{0,1728\}$. Let $Y_1(n)^\circ$ be the open subscheme obtained by removing the fibers above $j = 0,1728$, and let $\mathbb{A}_j^{1\circ}$ be the $j$-line minus $0,1728$.

One may consider the fiber products $$A := Y_1(5)^\circ\times_{\mathcal{M}_{1,1}^\circ}Y_1(7)^\circ,\qquad B := Y_1(5)^\circ\times_{\mathbb{A}_j^{1\circ}}Y_1(7)^\circ$$

Both $A,B$ are schemes. $Y_1(7)^\circ$ is finite etale over $\mathcal{M}_{1,1}^\circ$ of degree $[SL_2(\mathbb{Z}):\Gamma_1(7)] = 48$, in the sense that the fiber category above a geometric point $\text{Spec }k\rightarrow\mathcal{M}_{1,1}^\circ$ is a groupoid with 48 objects, but with isomorphisms between them (48 should be the degree according to the definition of the Galois category of an algebraic stack). Thus, since we're staying away from $j = 0,1728$, the resulting scheme theoretic fiber should have degree 24 over $\text{Spec }k$. For the same reason, $A$ should be finite etale over $Y_1(5)^\circ$ of degree 24.

Similarly, $Y_1(7)^\circ$ is finite etale over $\mathbb{A}_j^{1\circ}$, and so the pullback $B$ is finite etale over $Y_1(5)^\circ$, also of degree 24.

The universal property of fiber products provides a map $A\rightarrow B$ (commuting with the finite etale maps to $Y_1(5)^\circ$). This map must then also be finite etale, and by comparing degrees, must be an isomorphism.

Is this correct? (That $A\cong B$?)

In particular, this would seem to say that if we have an elliptic curve $E_1/K$ with a point $P_1$ of order 5, and an elliptic curve $E_2/K$ with a point $P_2$ of order 7, such that $j(E_1) = j(E_2) \ne 0,1728$, then this pair $((E_1,P_1),(E_2,P_2))$ would correspond to a $K$-point of $B$, and hence a $K$-point of $A$, but by definition of the fiber product of stacks, giving such a $K$-point of $A$ would imply that in fact $E_1\cong E_2$ over $K$ - which says more than just $j(E_1) = j(E_2)$ - ie, $E_1$ is not a twist of $E_2$.

If so this seems quite striking. In particular, this would seem to imply that if an elliptic curve $E/K$ with automorphism group $\{\text{id},[-1]\}$ has a $K$-point of order $n\ge 4$ (so that $Y_1(n)$ is a fine moduli scheme), then none of the twists of $E/K$ can have $K$-points of order $\ge 4$. Ie, amongst the set of all twists of $E$, at most one can admit a $K$-rational point of order $\ge 4$.