Quadratic points on bielliptic modular curves

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman Siksek [Math. Comp. 88 (2019), 2461–2484], Box 90 (2021), 321–343] described all the quadratic points on modular curves of genus 2 ≤ g ( X 0 n stretchy="false">) 5 2\leq g(X_0(n)) \leq 5 . Since hyperelliptic alttext="upper right-parenthesis"> encoding="application/x-tex">X_0(n) are alttext="less-than-or-equal-to encoding="application/x-tex">\leq as a curve can have infinitely many only if it is either 1"> 1 1 , or bielliptic, question describing bielliptic naturally arises; this has recently also been posed by Mazur. We answer Mazur’s completely describe for which not done already. The values alttext="n"> encoding="application/x-tex">n that we deal with alttext="n equals 60"> = 60 encoding="application/x-tex">n=60 alttext="62"> 62 encoding="application/x-tex">62 alttext="69"> 69 encoding="application/x-tex">69 alttext="79"> 79 encoding="application/x-tex">79 alttext="83"> 83 encoding="application/x-tex">83 alttext="89"> 89 encoding="application/x-tex">89 alttext="92"> 92 encoding="application/x-tex">92 alttext="94"> 94 encoding="application/x-tex">94 alttext="95"> 95 encoding="application/x-tex">95 alttext="101"> 101 encoding="application/x-tex">101 alttext="119"> 119 encoding="application/x-tex">119 alttext="131"> 131 encoding="application/x-tex">131 ; up to alttext="11"> 11 encoding="application/x-tex">11 find exceptional these show they correspond CM elliptic curves. two main methods use Box’s relative symmetric Chabauty method an application moduli description alttext="double-struck Q"> Q encoding="application/x-tex">\mathbb {Q} -curves degree alttext="d"> d encoding="application/x-tex">d independent isogeny alttext="m"> m encoding="application/x-tex">m reduces problem finding rational several quotients