Gonality of an Algebraic Curve

The gonality of an algebraic curve over a field is the least degree of a nonconstant morphism , where is the projective line and is smooth and projective, and remains irreducible over an algebraic closure of . Equivalently, it is the least degree of a nonconstant element of its function field (Derickx and van Hoeij 2014). The field is part of the definition, since extending it can change the available maps. This curve invariant motivates the analogous gonality of a graph.

Maiti (2026) determined

where is the modular curve parametrizing elliptic curves with a point of order 42. The lower bound uses exhaustive computations in the function field over . Together with the finiteness of , the rational points of its Jacobian variety, this shows that has only finitely many points of degree at most 11 over and infinitely many of degree 12 over .