Let be the maximal totally real subfield of , the cyclotomic field of -nd roots of unity. Let be the quaternion algebra over ramified exactly at the unique prime above and 7 of the real places of . Let be a maximal order in , and the Shimura curve attached to . Let , where is the unique Atkin–Lehner involution on . We show that the curve has several striking features. First, it is a hyperelliptic curve of genus , whose hyperelliptic involution is exceptional. Second, there are Weierstrass points on , and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension of class number contained in , the cyclotomic field of -th roots of unity. Third, the normal closure of the field of -torsion of the Jacobian of is the Harbater field , the unique Galois number field unramified outside and , with Galois group . In fact, the Jacobian has the remarkable property that each of its simple factors has a -torsion field whose normal closure is the field . Finally, and perhaps the most striking fact about , is that it is also hyperelliptic over .
"An intriguing hyperelliptic Shimura curve quotient of genus 16." Algebra Number Theory 14 (10) 2713 - 2742, 2020. https://doi.org/10.2140/ant.2020.14.2713