95440 Bayreuth, Germany
On November 9, 2013, in the course of a systematic search for curves of genus 2 defined over Q with a large hyperelliptic torsion packet, I discovered the following curve.
It has exactly 34 points in the hyperelliptic torsion packet, with x-coordinates in the following list.
infinity, 0, roots of x6 + 130 x3 + 13, roots of x12 - 91 x9 - 273 x6 - 1183 x3 + 169.
The differences of these points give 576 distinct points on the Jacobian of order dividing 48.
Bjorn Poonen was so kind to verify that there are no further points in the torsion packet using the program described in his paper Computing torsion points on curves (Experiment. Math. 10 (2001), no. 3, 449–465). The previous record (even for a curve over C) was 22. Another paper of Bjorn's (Genus-two curves with 22 torsion points, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 573–576) proves that there are infinitely many curves over R with 22 points (or more) in their hyperelliptic torsion packets.