For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

Two integral structures on the $\mathbb{Q}$-vector space of modular forms of weight two on $X_0(N)$ are compared at primes $p$ dividing $N$ at most once. When $p=2$ and $N$ is divisible by a prime that is $3$ mod $4$, this comparison leads to an algorithm for computing the space of weight one forms mod $2$ on $X_0(N/2)$. For $p$ arbitrary and $N>4$ prime to $p$, a way to compute the Hecke algebra of mod $p$ modular forms of weight one on $\varGamma_1(N)$ is presented, using forms of weight $p$, and, for $p=2$, parabolic group cohomology with mod $2$ coefficients. Appendix A is a letter of October 1987 from Mestre to Serre in which he reports on computations of weight one forms mod $2$ of prime level. Appendix B, by Wiese, reports on an implementation for $p=2$ in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.

In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.

This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.

All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.

1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.