After discarding preperiodic points, the question is: how small can the canonical height of z (for the map z2 - c) be, relative to the (standard) height of c?
The following two comma-separated text files list all such pairs (c,z) where the ratio of the two heights is less than about 0.037:
n not divisible by 4 n divisible by 4
Reading the data files:
Each line of each data file is in the format
c,z,[canonical height h_c(z) of z],[height ratio h_c(z)/h(c)]
Here is a sample entry, which appears in the ``n divisible by 4'' file:
This means the point 10541/21840 has canonical height about 0.30762 for the map z2 - 1013082841/476985600. Dividing this height by h(1013082841/476985600)= log(1013082841) gives the height ratio of about 0.014835.