We prove that there are \(\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}\) imaginary quadratic fields k with discriminant \(|d_k|\le X\) and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound \(\gg X^{\frac{1}{4}}\) in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curve C over \(\mathbb {Q}\) such that C has a rational Weierstrass point and the Jacobian of C has a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curve C and a quantitative result of Kulkarni and the second author.
