On PGL2(F7) and PSL2(F7) number fields ramified at a single prime

We present new examples of ${\mathrm{PGL}}_2\left(F_7\right)$ and ${\mathrm{PSL}}_2\left(F_7\right)$ number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's ${\mathrm{PGL}}_2\left(F_7\right)$ polynomial, (ii) Modular method: computation of Katz modular forms of weight one over ${\bar{F}}_7$ with prime level, and (iii) Searching for polynomials with prescribed ramification.

Method (i) quickly generates many ${\mathrm{PGL}}_2\left(F_7\right)$ number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of ${\mathrm{PGL}}_2\left(F_7\right)$ or ${\mathrm{PSL}}_2\left(F_7\right)$ number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.