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\title[Points of prime order over number fields of degree 9 to 22]{Points of prime order on elliptic curves over number fields of degree $9$ to $22$}

\author{Filip Najman}
\address{University of Zagreb Faculty of Science, Department of Mathematics, Bijeni\v cka cesta 30, 10000 Zagreb, Croatia}
\email{fnajman@math.hr}
\thanks{The author was supported by the Croatian Science Foundation under the project no.\ IP-2022-10-5008, by the project ``Implementation of cutting-edge research and its application as part of the Scientific Center of Excellence for Quantum and Complex Systems, and Representations of Lie Algebras'', PK.1.1.10.0004, co-financed by the European Union through the European Regional Development Fund -- Competitiveness and Cohesion Programme 2021--2027, and by the European Union -- NextGenerationEU through the National Recovery and Resilience Plan 2021--2026, via an institutional grant of the University of Zagreb Faculty of Science, IK IA 1.1.3, Impact4Math.}
\subjclass[2020]{11G05, 11G18, 14G05}
\keywords{elliptic curves, torsion, modular curves, number fields}

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\begin{document}

\begin{abstract}
Let $S(d)$ be the set of primes $p$ for which there exists an elliptic curve over a number field of degree $d$ with a point of order $p$. The sets $S(d)$ are known for $d\leq 8$. We determine $S(d)$ for all $9\leq d\leq 22$. For example, the largest prime in $S(22)$ is $67$, and $S(16)$ consists of the primes up to $37$. To reduce the problem to finitely many primes we prove a version of Oesterl\'e's bound which uses reduction modulo $2$ instead of modulo $3$, and which is much smaller for $d\geq 13$. The remaining primes are dealt with using results of Derickx and Stoll and extensive computations with the modular curves $X_1(p)$ over $\F_2$ for $p\leq 139$.
\end{abstract}

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