According to [this reference][1] an elliptic curve is $F$-pure if and only if the $F$-pure threshold of its defining ideal is $1$. Does there exist an $F$-pure local ring $R=A/\mathfrak{a}$ such that $\text{fpt}(\mathfrak{a})\neq 1$?


  [1]: http://www.math.ias.edu/~bhattb/math/cyfthreshold.pdf