Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn varying over all possible $E$. My question is the following: does there exist an integer $N=N_d$ such that $N \geq \#|E(K_{\mathrm{tors}})|$ for any $K, E$ and fixed $d$? If this is not known to be the case, then what is the general consensus regarding the number of $K$-rational torsion points of $E$? If none of this is known, is there a record for the largest known torsion order? As of right now I'm curious for $d=2$ and non-CM elliptic curves $E(K)$, but any information regarding a specific $d > 1$ is helpful.

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    $\begingroup$ Yes, such N exists for each d; Loïc Merel proved this (with an explicit albeit large N) in a paper published by Inventiones mathematicæ in 1996. For $d=2$ The problem is completely solved. en.wikipedia.org/wiki/Torsion_conjecture $\endgroup$ Nov 10, 2022 at 4:47
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    $\begingroup$ @NoamD.Elkies Your comment should be an answer (so that this question can be closed). $\endgroup$
    – GH from MO
    Nov 10, 2022 at 7:21
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    $\begingroup$ What does "over the field $K$ with coefficients in $K$" mean? I know what "over the field $K$" means (i.e. the curve is defined over $K$ and the basepoint is a $K$-point) — I would rather have written "an elliptic curve $E$ over the field $K$", with $E(K)$ rather denoting the group of $K$-points. $\endgroup$
    – YCor
    Nov 10, 2022 at 7:37
  • $\begingroup$ @NoamD.Elkies Thanks, this is a great answer. This question should be closed now. $\endgroup$ Nov 11, 2022 at 1:12
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    $\begingroup$ @GHfromMO I didn't yet formulate my comment as an answer because I wanted to first look up the list for $d=2$ and to credit it properly. $\endgroup$ Nov 11, 2022 at 1:27


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