We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
Matthias Schütt. Tetsuji Shioda. "An interesting elliptic surface over an elliptic curve." Proc. Japan Acad. Ser. A Math. Sci. 83 (3) 40 - 45, March 2007. https://doi.org/10.3792/pjaa.83.40