Despite being the widely-used gold standard for linking common genetic variations to phenotypes and disease, genome-wide association studies (GWAS) suffer major limitations, partially attributable to the reliance on simple, typically linear, models of genetic effects. More elaborate methods, such as epistasis-aware models, typically struggle with the scale of GWAS data. In this paper, we build on recent advances in neural networks employing Transformer-based architectures to enable such models at a large scale. As a first step towards replacing linear GWAS with a more expressive approximation, we demonstrate prediction of gout, a painful form of inflammatory arthritis arising when monosodium urate crystals form in the joints under high serum urate conditions, from Single Nucleotide Variants (SNVs) using a scalable (long input) variant of the Transformer architecture. Furthermore, we show that sparse SNVs can be efficiently used by these Transformer-based networks without expanding them to a full genome. By appropriately encoding SNVs, we are able to achieve competitive initial performance, with an AUROC of 83% when classifying a balanced test set using genotype and demographic information. Moreover, the confidence with which the network makes its prediction is a good indication of the prediction accuracy. Our results indicate a number of opportunities for extension, enabling full genome-scale data analysis using more complex and accurate genotype-phenotype association models.