With Michael Gill I enumerated the P1Fs of K_{16}. It turns out that there are 3155 of them, up to isomorphism. This includes 89 with non-trivial automorphism group. You can download the P1Fs here. They are written using vertices a,b,...,p. Each line lists the 15 factors that comprise the P1F; each factor is given as 8 edges concatenated without spaces. Every P1F includes the factors abcdefghijklmnop and apbcdefghijklmno, and is the lexicographically least representative of its isomorphism class under this constraint. The list is sorted in decreasing order of the size of the automorphism group. Representatives with the same sized automorphism groups are sorted lexicographically.
Note that this page will be periodically updated and the date on which each new result was added is given at the end of each line.
p=101, q=1030301, ζ(x)=x^{3}+x+3, c=[813092,759910,233271,3] (24 Mar 2007)
p=103, q=1092727, ζ(x)=x^{3}+x+4, c=[828376,896] (31 May 2006)
p=107, q=1225043, ζ(x)=x^{3}+x+9, c=[1107573,151] (31 May 2006)
p=109, q=1295029, ζ(x)=x^{3}+x+6, c=[271574,645911,1082655,4] (17 Apr 2007)
p=127, q=2048383, ζ(x)=x^{3}+x+15, c=[840749,23] (24 Mar 2007)
p=131, q=2248091, ζ(x)=x^{3}+x+3, c=[2096100,298] (31 May 2006)
p=139, q=2685619, ζ(x)=x^{3}+x+7, c=[436598,2118] (31 May 2006)
p=149, q=3307949, ζ(x)=x^{3}+x+14, c=[1861398,3141536,1357853,1] (17 Apr 2007)
p=151, q=3442951, ζ(x)=x^{3}+x+5, c=[1492322,66] (31 May 2006)
p=163, q=4330747, ζ(x)=x^{3}+x+4, c=[2015256,4602] (24 Mar 2007)
p=167, q=4657463, ζ(x)=x^{3}+x+3, c=[3183263,109] (31 May 2006)
p=179, q=5735339, ζ(x)=x^{3}+x+4, c=[2740965,1219] (31 May 2006)
p=191, q=6967871, ζ(x)=x^{3}+x+3, c=[4789910,1160] (24 Mar 2007)
p=199, q=7880599, ζ(x)=x^{3}+x+13, c=[3457494,2368] (24 Mar 2007)
p=211, q=9393931, ζ(x)=x^{3}+x+24, c=[5457264,1168] (24 Mar 2007)
p=223, q=11089567, ζ(x)=x^{3}+x+9, c=[4722613,4305] (24 Mar 2007)
p=227, q=11697083, ζ(x)=x^{3}+x+9, c=[9051956,1442] (24 Mar 2007)
p=239, q=13651919, ζ(x)=x^{3}+x+11, c=[1597504,5918] (24 Mar 2007)
p=251, q=15813251, ζ(x)=x^{3}+x+7, c=[9285089,11965] (24 Mar 2007)
p=263, q=18191447, ζ(x)=x^{3}+x+8, c=[8313030,2840] (24 Mar 2007)
p=271, q=19902511, ζ(x)=x^{3}+x+4, c=[6563520,170] (24 Mar 2007)
p=283, q=22665187, ζ(x)=x^{3}+x+24, c=[2245440,3574] (24 Mar 2007)
p=19, q=2476099, ζ(x)=x^{5}+x+9, c=[949007,791] (24 Mar 2007)
p=23, q=6436343, ζ(x)=x^{5}+x+3, c=[1045440,7580] (24 Mar 2007)