This page has links to all the data concerning

The data concerns pairs (

The following text files list all such preperiodic
pairs (*x*,*f*) our search found, grouped by the
length of the cycle period. Period 2, Length 6 had
far too many examples and is omitted.

Period 2, Length 8 Period 3, Length 8

Period 1, Length 7 Period 2, Length 7 Period 3, Length 7 Period 4, Length 7 Period 5, Length 7 Period 6, Length 7 Period 7, Length 7

Period 1, Length 6 (Per 2, Len 6 omitted) Period 3, Length 6 Period 4, Length 6 Period 5, Length 6 Period 6, Length 6

The following text file lists all
such pairs (*x*,*f*) where the ratio of the
two heights is less than about 0.002.
Within the file, pairs appear in order of increasing height ratio.

The following text file is the original raw data, i.e., the union of the above files, but in the order the search found them, i.e., unsorted.

**Reading the data files:**

Here is a sample entry from the Period 4 file:

`
phi(w) = (-1925*w^2 + 5071*w - 3146)/(-1925*w^2 + 5305*w + 7150)
`

`
inf --> 1 --> 0 --> -11/25 --> -22/17 --> 22/5 --> 187/70 --> -11/25
`

The first line gives the formula for the function; the second gives the forward orbit of infinity (inf). Note the repetition of -11/25 in this example, showing the cycle of period 4.

Here is a sample entry from the Small file:

`
phi(w) = (-60*w^2 + 24*w + 36)/(-60*w^2 + 143*w - 6)
`

`
inf --> 1 --> 0 --> -6 --> 3/4 --> 3/10 --> 6/5 --> -3/11 --> -48/95 --> -78/853
`

`
hhat(inf) = 0.011308589139336591803864543246008924166
`

`
h(phi) = 12.056347989725183163302238437525483561
`

`
hhat(inf)/h(phi) = 0.00093797799706628772830489630836933015025
`

Again, the first line gives a formula for the function, and the second gives the first 10 iterates in the (non-preperiodic) forward orbit of infinity (inf).

The third line gives an approximation of the canonical height of infinity; the fourth gives the height of phi as a point in the moduli space; and the fifth is the quotient of these two heights.

**NOTE**: the decimal approximation for hhat(inf), and hence
also the quotient hhat(inf)/h(phi), are only reliable to
four or five places past the decimal point. They are simply
the raw outputs from PARI, which is why so many digits appear
in the expansion, even though most of the digits that appear
are completely unreliable. However, the first four and usually
five digits after the decimal point are accurate.