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](https://arxiv.org/pdf/math/0201021.pdf)). 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 $\ge 4$.



Torsion points on twists of elliptic curves and products of fine modular curves over $\mathcal{M}_{1,1}$ vs over the $j$-line
I think your degree calculation is bogus. You write:



> 
> $Y\_1(7)^\circ$ 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 [...] The resulting scheme theoretic fiber should have degree 24 over $\text{Spec }k$.
> 
> 
> 


But the map $Y\_1(7)^\circ \to \mathcal{M}\_{1,1}^\circ$ is representable, and this means in particular that the fiber category over $\text{Spec }k$ (or over any scheme for that matter) is an algebraic space, not a stack. So the only isomorphisms in your 48-object category are identities, and the "scheme theoretic fiber" should have degree 48 over $\text{Spec }k$.


