**Michael Stoll**

Mathematisches Institut

Universität Bayreuth

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 *x*^{6} + 130 *x*^{3} + 13,
roots of *x*^{12} - 91 *x*^{9}
- 273 *x*^{6} - 1183 *x*^{3} + 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.