List of toric log del Pezzo surfaces of Picard number one with index 16

Caution. There are two toric log del Pezzo surfaces of index 16 with singularity type [2, 2, 3, 2, 5],[2, 2, 2, 2, 4],[2, 2, 2, 3, 4]. It has the following cascade.
[2, 2, 2, 2, 4], [2, 2, 2, 3, 4], [2, 2, 3, 2, 5] \(\rightarrow\) [2, 2, 2, 4], [2, 2, 2, 3, 3], [2, 2, 3, 2, 5] \(\rightarrow\) [2, 2, 4], [2, 2, 2, 3, 2], [2, 2, 3, 2, 5] \(\rightarrow\) [2, 2, 3], [2, 2, 2, 3, 2], [2, 3, 2, 5] \(\rightarrow\) [2, 2, 2], [2, 2, 2, 3, 2], [3, 2, 5] \(\rightarrow\) [2, 2], [2, 2, 5], [2, 2, 2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 4], [2, 2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 3], [2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 2], [2, 3] \(\rightarrow\) [2, 2], [2, 2], [2, 2]
[2, 2, 2, 2, 4], [2, 2, 2, 3, 4], [2, 2, 3, 2, 5] \(\rightarrow\) [2, 2, 2, 4], [2, 2, 2, 3, 4], [2, 2, 3, 2, 4] \(\rightarrow\) [2, 2, 4], [2, 2, 3, 2, 3], [2, 2, 2, 3, 4] \(\rightarrow\) [2, 4], [2, 2, 3, 2, 2], [2, 2, 2, 3, 4] \(\rightarrow\) [2, 3], [2, 2, 3, 2, 2], [2, 2, 3, 4] \(\rightarrow\) [2, 2], [2, 2, 3, 2, 2], [2, 3, 4] \(\rightarrow\) [2, 2], [2, 2, 3, 2], [2, 3, 3] \(\rightarrow\) [2, 2], [2, 2, 3], [2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 2], [2, 3] \(\rightarrow\) [2, 2], [2, 2], [2, 2]

47 singularity types

Singularity types \(K^2\) Orders of local
fundamental groups		 Local indices Index Cascade
[32] 289/8 32 16 16 cascade
[2], [2], [2, 9, 2] 81/8 2, 2, 32 1, 1, 16 16 cascade
[2], [2, 2, 2, 2, 2], [2, 6, 6] 27/4 2, 6, 64 1, 1, 16 16 cascade
[4], [2, 2, 2], [2, 2, 2, 3, 4] 25/8 4, 4, 32 2, 1, 16 16 cascade
[2, 2], [2, 2, 2, 2, 2, 2], [3, 2, 7] 21/8 3, 7, 32 1, 1, 16 16 cascade
[2, 2, 2], [3, 2, 2, 2, 3], [2, 6, 2, 2, 4] 15/8 4, 20, 96 1, 2, 16 16 cascade
[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [6, 3, 6] 27/8 6, 6, 96 1, 1, 16 16 cascade
[2, 3, 2], [2, 2, 2, 2, 2, 2, 2], [3, 2, 7] 9/8 8, 8, 32 4, 1, 16 16 cascade
[2, 3, 2], [2, 2, 4, 2, 2], [3, 2, 3, 2, 4] 3/4 8, 24, 64 4, 4, 16 16 cascade
[3, 3], [2, 2, 2, 3, 4], [2, 2, 3, 3, 2, 2] 5/8 8, 32, 40 2, 16, 4 16 cascade
[8], [3, 3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2] 9/8 8, 8, 32 4, 2, 16 16 cascade
[2, 2, 2], [2, 9, 2], [3, 2, 2, 2, 2, 2, 2, 2, 3] 9/8 4, 32, 36 1, 16, 2 16 cascade
[2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 3, 4], [5, 2, 5] 5/8 8, 32, 40 1, 16, 4 16 cascade
[2, 2, 2, 2, 4], [2, 2, 2, 3, 4], [2, 2, 3, 2, 5] 3/8 16, 32, 48 8, 16, 4 16 cascade
[2, 2, 2, 2, 2, 2, 2], [3, 2, 2, 2, 2, 3], [2, 6, 6] 3/4 8, 24, 64 1, 2, 16 16 cascade
[2, 2, 2, 3, 4], [2, 2, 2, 3, 4], [2, 2, 3, 2, 2, 5] 1/4 32, 32, 64 16, 16, 4 16 cascade
[2, 2, 3, 5], [2, 2, 2, 2, 2, 4, 2], [3, 2, 3, 2, 4] 1/4 32, 32, 64 4, 8, 16 16 cascade
[2, 2, 4, 2, 2], [3, 2, 2, 2, 2, 3], [2, 6, 2, 2, 4] 3/8 24, 24, 96 4, 2, 16 16 cascade
[2, 3, 2], [2, 9, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 3] 5/8 8, 32, 40 4, 16, 2 16 cascade
[2, 2, 6], [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2], [3, 6, 3] 3/8 16, 32, 48 4, 16, 8 16 cascade
[2, 2, 6], [3, 2, 7], [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2] 3/8 16, 32, 48 4, 16, 8 16 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [17, 17] 153/8 17, 288 1, 16 16 cascade
[2, 2, 2, 2, 2, 4, 2], [3, 2, 2, 2, 2, 2, 2, 3], [2, 6, 6] 1/4 32, 32, 64 8, 2, 16 16 cascade
[2, 5, 2], [2, 9, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 3/8 16, 32, 48 8, 16, 2 16 cascade
[3, 2, 2, 3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2], [2, 3, 10] 3/8 16, 32, 48 2, 16, 8 16 cascade
[3, 2, 7], [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2], [2, 3, 5, 3] 1/4 32, 32, 64 16, 16, 8 16 cascade
[8], [5, 5], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2] 3/4 8, 24, 64 4, 4, 16 16 cascade
[2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 5, 18] 45/8 2, 18, 160 1, 1, 16 16 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [3, 2, 7], [2, 3, 10] 3/8 16, 32, 48 1, 16, 8 16 cascade
[3, 2, 7], [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2], [2, 2, 2, 2, 2, 2, 10] 1/4 32, 32, 64 16, 16, 8 16 cascade
[3, 11], [2, 2, 3, 5], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2] 1/4 32, 32, 64 8, 4, 16 16 cascade
[2, 9, 2], [2, 9, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 1/4 32, 32, 64 16, 16, 2 16 cascade
[2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 22] 11/4 2, 22, 64 1, 1, 16 16 cascade
[2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [6, 2, 18] 9/4 6, 18, 192 1, 1, 16 16 cascade
[4], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 20] 15/8 4, 20, 96 2, 1, 16 16 cascade
[3, 2, 2, 2, 3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 5, 18] 5/8 20, 20, 160 2, 1, 16 16 cascade
[3, 3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 22] 3/4 8, 24, 64 2, 1, 16 16 cascade
[5, 5], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 20] 3/8 24, 24, 96 4, 1, 16 16 cascade
[32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 33/8 32, 33 16, 1 16 cascade
[2], [32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 17/8 2, 32, 34 1, 16, 1 16 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [17, 2, 17] 17/8 17, 17, 544 1, 1, 16 16 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [17, 17] 9/8 18, 18, 288 1, 1, 16 16 cascade
[3, 11], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 22] 1/4 32, 32, 64 8, 1, 16 16 cascade
[4], [32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 9/8 4, 32, 36 2, 16, 1 16 cascade
[8], [32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 5/8 8, 32, 40 4, 16, 1 16 cascade
[16], [32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 3/8 16, 32, 48 8, 16, 1 16 cascade
[32], [32], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 1/4 32, 32, 64 16, 16, 1 16 cascade