## Richard Borcherds' Complete List of 25-Dimensional Unimodular Lattices

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, RWTH Aachen university (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, (njasloane@gmail.com).

## The Complete List of 25-Dimensional Unimodular Lattices

Richard Borcherds
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge, England
Email address R.E.Borcherds@pmms.cam.ac.uk

Taken from R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, University of Cambridge, 1984.

• This is a table of the 25-dimensional unimodular lattices.

### Table -4.  The norm -4 vectors of II25,1.

There is a natural 1:1 correspondence between the elements of the following sets:

(1)
Orbits of norm -4 vectors u in II25,1 under Aut(II25,1).
(2)
Orbits of norm -4 vectors in the fundamental domain D of II25,1 under Aut(D).
(3)
Orbits of norm -1 vectors v of I25,1 under Aut(I25,1).
(4)
25 dimensional unimodular positive definite lattices L.
(5)
Unimodular lattices L1 of dimension at most 25 with no vectors of norm 1.
(6)
25 dimensional even lattices L2 of determinant 4.
L1 is the orthogonal complement of the norm 1 vectors of L, L2 is the lattice of elements of L of even norm, L2 is isomorphic to u^, and L is isomorphic to v^. Table -4 lists the 665 elements of any of these sets.

The height t is the height of the norm -4 vector u of D, in other words -(u,w) where w is the Weyl vector of D. The things in table -4 are listed in increasing order of their height.

Dim is the dimension of the lattice L1. A capital E after the dimension means that L1 is even.

The column ``roots'' gives the Dynkin diagram of the norm 2 vectors of L2 arranged into orbits under Aut(L2).

``Group'' gives the order of the subgroup of Aut(D) fixing u. The group Aut(L) @ Aut(L2) is of the form 2×R.G where R is the group generated by the reflections of norm 2 vectors of L, G is the group described in the column ``group'', and 2 is the group of order 2 generated by -1. If dim(L1) £ 24 then Aut(L1) is of the form R.G where R is the reflection group of L1 and G is as above.

For any root r of u^ the vector u+r is a norm -2 vector of II25,1. This vector u can be found as follows. Let X be the component of the Dynkin diagram of u^ to which u belongs and let h be the Coxeter number of X. Then r+u is conjugate to a norm -2 vector of II25,1 in D of height t-h+1 (or t-h if the entry under ``Dim'' is 24E) whose letter is the letter corresponding to X in the column headed ``norm -2's''. For example let u be the vector of height 6 and root system a22a110. Then the norm -2 vectors corresponding to roots from the components a2 or a1 have heights 6-3+1 and 6-2+1 and letters a and b, so they are the vectors 4a and 5b of table -2.

If dim(L1) £ 24 then the column ``neighbors'' gives the two even neighbors of L1+I24-dim(L1). If dim(L1) £ 23 then both neighbors are isomorphic so only one is listed, and if L1 is a Niemeier lattice then the neighbor is preceded by 2 (to indicate that the corresponding norm 0 vector is twice a primitive vector). If the two neighbors are isomorphic then there is an automorphism of L exchanging them.

 Height Dim Roots Group norm -2's Neighbors 1 24 E 8315553613086720000 2L 2 23 a12 84610842624000 a L 2 24 1002795171840 L A124 3 25 a12 88704000 a 4 24 a18 20643840 a A124 A124 4 25 a22 26127360 a 4 25 a16 138240 a 5 24 a112 190080 a A124 A212 5 25 a2a17 5040 aa 5 25 a110 1920 a 6 23 a116a12 645120 ca A124 6 24 a22a110 5760 ab A212 A212 6 24 a116 43008 c A124 A38 6 25 a3a18 21504 ac 6 25 a22a18 128 ab 6 25 a2a110a1 120 abc 6 25 a18a16 1152 bc 7 24 a24a18 384 bb A212 A38 7 24 E a124 244823040 a 2A124 7 25 a25a13 720 bb 7 25 a3a2a19 72 acb 7 25 a24a14a12 24 bba 7 25 a3a112 1440 ab 7 25 a23a16a13 12 bbb 7 25 a22a112 144 cb 8 22 a3a122 887040 ae A124 8 23 a26a16a12 1440 bda A212 8 24 a32a112 768 cc A38 A38 8 24 a3a24a16 96 bbb A38 A38 8 24 a28 672 a A38 A38 8 24 a26a16 240 bd A212 A46 8 24 a124 138240 e A124 D46 8 25 a4a112 1440 ad 8 25 a3a24a14 16 bbb 8 25 a32a18a12 64 cbc 8 25 a3a23a16a1 12 bbbc 8 25 a3a23a13a13a1 6 bbbbd 8 25 a24a22a14 8 bab 8 25 a3a22a14a14a12 16 bbcbd 8 25 a24a2a14a12a1 8 bbbdb 8 25 a24a18a12 48 bdc 8 25 a3a115a1 720 cce 9 24 a32a24a14 16 bbb A38 A46 9 24 a28a14 384 dc A212 A54D4 9 25 a4a23a16a1 12 bdbb 9 25 a32a24a12 16 bba 9 25 a32a22a2a14a1 4 bbcba 9 25 a3a3a22a2a12a12a1 2 bbbbbbb 9 25 a3a26a12 6 abb 9 25 a32a22a14a14 8 bcbb 9 25 a3a24a2a14a1 8 bbbbc 9 25 a3a22a22a2a12a12a1 2 bbbdbbb 10 22 a3a210 2880 ac A212 10 23 a34a18a12 384 cfa A38 10 23 a33a24a12a12 48 bbea A38 10 24 a34a22a12 32 bab A46 A46 10 24 a4a3a24a14 16 bdbc A46 A46 10 24 a32a3a24a12 16 bbbe A38 A54D4 10 24 a34a14a14 48 cdf A38 A54D4 10 24 a34a18 384 cd A38 D46 10 24 E a212 190080 a 2A212 10 25 a35 1920 c 10 25 d4a24a16 144 bcd 10 25 d4a3a112 576 ced 10 25 a5a115 720 cf 10 25 a4a3a24a12 8 bdbb 10 25 a33a3a2a13 6 bcbb 10 25 a4a3a22a2a12a12a1 2 bdbbbce 10 25 a33a3a14a12 24 bcce 10 25 a32a32a14a12 16 cbbd 10 25 a4a26a12 6 abe 10 25 a4a3a22a14a12a12 4 bdbccf 10 25 a32a3a22a2a12a1 2 bbbbbc 10 25 a32a3a22a2a1a1a1 2 bbbadbc 10 25 a4a25a15 10 bbc 10 25 a32a26 48 da 10 25 a32a24a22 8 bba 10 25 a33a22a13a12a1 12 bbdcf 10 25 a3a3a3a22a12a1a1a1a1 2 cbbbcebfd 10 25 a3a3a22a22a2a12a1 2 bbbbbee 10 25 a32a24a14a12 8 dbcf 10 25 a33a112 48 cf 10 25 a3a26a2a13 12 dbce 11 24 a4a32a3a22a12 4 bbbbb A46 A54D4 11 24 a42a24a14 16 dca A46 A54D4 11 24 a36 240 a A46 D46 11 24 a34a24 24 be A38 A64 11 25 a5a24a2a14 8 befb 11 25 d4a3a24a14 8 bcda 11 25 a4a32a3a2a12a1 2 bbbcba 11 25 a42a22a2a14a1 4 dcbbb 11 25 a4a32a24 4 bbb 11 25 a4a33a16 6 cbb 11 25 a4a32a22a2a12a1 2 bbbfab 11 25 a4a3a3a2a2a2a1a1a1 1 bbbbccbba 11 25 a4a3a3a2a2a2a1a1a1 1 bbbbecbbb 11 25 a34a3a14 8 bab 11 25 a32a32a22a2a1 2 bbbbb 11 25 a32a3a3a22a2a1 2 bbabda 11 25 a4a3a22a22a2a12a1 2 bbeccba 11 25 a32a32a22a14 4 bbdb 12 22 a36a3a12 96 dag A38 12 23 a42a32a22a12a12 8 bcbha A46 12 23 a4a35a12 40 aba A46 12 24 d4a4a26 24 dca A54D4 A54D4 12 24 d4a34a14 32 cde A54D4 A54D4 12 24 a5a33a16a1 24 cfec A54D4 A54D4 12 24 a42a32a3a12 8 bbbd A54D4 A54D4 12 24 a5a32a24a1 16 bebf A54D4 A54D4 12 24 d4a34a14 48 cdc D46 A54D4 12 24 d42a116 1152 eb D46 D46 12 24 a42a32a22a12 4 bcbh A46 A64 12 24 a34a32a14 32 fdg A38 A72D52 12 25 a5a32a3a14a1 8 cefdc 12 25 a5a32a22a2a1a1 2 bebbdh 12 25 d4a32a3a22a12 4 bddac 12 25 d4a4a24a14 8 dcae 12 25 a5a32a22a12a12a1 4 bfbdhf 12 25 a5a3a3a22a12a1a1a1 2 bfebhdee 12 25 a42a3a3a2a12a1 2 bbcade 12 25 a4a4a3a3a2a1a1a1 1 bbccbddd 12 25 a42a3a24 4 bbb 12 25 a42a32a14a12 4 bche 12 25 d4a32a24a1a1 8 bdaeg 12 25 d4a33a16a1a1 12 cdegb 12 25 a4a32a32a2a1 2 abcbc 12 25 a42a3a22a2a1a1a1 2 bcbbfdh 12 25 a4a4a3a2a2a2a1a1a1 1 bbcabbhhd 12 25 a4a34a14 8 ach 12 25 a4a32a3a3a12a12 2 bbcfde 12 25 a4a33a23a1 6 bbag 12 25 a4a3a3a3a2a2a2a1 1 bbbebbah 12 25 a34a3a3a12 8 bfdc 12 25 a4a32a3a22a12a1a1 2 bccbhge 12 25 a4a32a3a22a12a12 2 bcfbhh 12 25 a32a32a3a22a12 4 edbbh 13 24 a44a14 24 cc A54D4 A64 13 24 a5a4a3a3a2a2a1 2 bebbddd A54D4 A64 13 24 a42a34 16 bb A46 A72D52 13 24 a42a4a3a22a12 4 ccahb A46 A72D52 13 24 E a38 2688 a 2A38 13 25 a6a26a13 12 dhd 13 25 a44a12 24 ca 13 25 a5a4a32a2a12 2 bfbdc 13 25 d4a4a33a12 6 bdac 13 25 d42a26 72 cc 13 25 a5a4a3a2a2a2a1a1 1 bebhdeda 13 25 a5a4a3a2a2a2a1a1 1 bebhdddd 13 25 d4a4a3a3a2a2a1a1 1 bdaaeccb 13 25 a42a4a3a22 2 bcbd 13 25 a4a4a4a3a2a1a1a1 1 bccbdbcd 13 25 a5a33a23 6 bbd 13 25 a5a32a3a22a2 2 abbhc 13 25 a5a4a22a22a2a12 2 behdfd 13 25 a5a32a3a22a12a1 2 bbbedd 13 25 a4a4a3a3a3a2a1 1 bcbbbdc 13 25 a4a4a3a3a3a2a1 1 bbbabdb 13 25 a43a23a13 6 cdb 13 25 d4a33a23a2 6 bacg 13 25 a42a3a3a22a2a1 2 ebaddd 13 25 a5a29 72 cf 14 21 d4a37 336 ag A38 14 22 a44a3a22 16 aab A46 14 23 a5a42a3a3a12a1 4 bbddaf A54D4 14 23 d4a43a22a12 12 caca A54D4 14 23 a5d4a33a13a12 12 dfega A54D4 14 23 a52a3a24a12 16 efda A54D4 14 23 d42a34a12 96 dcd D46 14 24 a5a43a13 12 cbe A64 A64 14 24 a52a32a22 8 eda A64 A64 14 24 a6a33a23 12 bhd A64 A64 14 24 d4a42a4a22 4 caac A54D4 A72D52 14 24 a5d4a32a3a12a1 4 dfecbg A54D4 A72D52 14 24 a52a32a12a12a12 8 febcg A54D4 A72D52 14 24 a5a42a3a3a1 4 bbddf A54D4 A72D52 14 24 d42a34 32 dc D46 A72D52 14 24 a43a33 12 bh A46 A83 14 24 a38 384 g A38 D64 14 25 d5a32a24a12 8 bgbb 14 25 d5a33a18 48 cgc 14 25 a6a32a3a22a1 2 bhhcf 14 25 a52a3a3a14 8 efee 14 25 a6a32a3a2a12a12 2 bhhdeg 14 25 d4a43a13a1 6 cabc 14 25 a5d4a3a3a22a1 2 deeccb 14 25 a5a5a3a22a12a1a1 2 efddefg 14 25 a5a42a3a2a12 2 bbfdf 14 25 a5a4a4a3a2a1a1 1 cbbdabe 14 25 d4a42a32a12 4 baeb 14 25 d42a32a3a14 8 dcbb 14 25 a5a42a3a12a12a1 2 cbhcee 14 25 a5d4a32a14a12a1 4 dfegcg 14 25 a5a4a3a3a3a2 1 bbhddc 14 25 a5a4a3a3a3a2 1 bbeddc 14 25 a5a42a22a2a12 2 cbddf 14 25 a44a22 8 ba 14 25 d4a4a4a3a2a2a1a1 1 caaecbbg 14 25 a5a4a3a3a3a1a1a1 1 bbhedbfg 14 25 a5a32a32a3a1 4 dddcg 14 25 a5a32a3a3a3a1 2 bheddf 14 25 a5a4a32a22a12a1 2 cbhdge 14 25 a42a4a32a2a1 2 bbdbf 14 25 a4a4a4a3a3a2a1 1 abbdhcf 14 25 a42a4a3a22a2a1 2 bahbdf 14 25 d4a34a3a14 8 fegg 15 24 a52a42a12 4 bdc A64 A72D52 15 24 a6a4a4a3a2a1a1 2 chhceaa A64 A72D52 15 24 a52a4a3a22 4 bfdf A54D4 A83 15 24 a44a32 16 hb A46 A92D6 15 25 d5a35 20 ab 15 25 d5a4a32a22a12 2 bgbba 15 25 a6a42a3a12a1 2 chdcb 15 25 a6a4a4a2a2a1a1a1 1 chhefacc 15 25 a5d4a42a12a1 2 abeab 15 25 a5a5a4a3a2a1 1 bbddea 15 25 a6a4a3a3a2a2a1 1 bhdcfga 15 25 a6a4a32a22a1 2 bhdfc 15 25 d42a42a22 4 acb 15 25 a52a4a3a12a12 2 bedcc 15 25 a5d4a4a3a2a2a1 1 abecbba 15 25 a52a32a3a12 2 bdac 15 25 a5a42a4a2a12 2 adfec 15 25 a5a42a4a2a12 2 addca 15 25 a5a4a4a4a2a1a1 1 bhddeca 15 25 a45 5 d 15 25 a5a4a4a3a3a2 1 bhdcab 15 25 d4a42a32a3 2 bccb 16 21 d4a45 40 ab A46 16 22 a52d4a32a3 8 ceba A54D4 16 22 a53a3a3a13 12 ecad A54D4 16 22 d44a3a16 144 bdc D46 16 23 a6a5a4a3a2a12a1 2 bhdecah A64 16 23 a53a4a12a1 6 daag A64 16 24 d5d4a34 16 dgb A72D52 A72D52 16 24 d5a5a32a3a12a1 4 fgbceb A72D52 A72D52 16 24 a52d42a12 16 cef A72D52 A72D52 16 24 a7a34a14 16 fge A72D52 A72D52 16 24 d5a4a42a22 4 cbba A72D52 A72D52 16 24 a6d4a42a2 4 ahcb A72D52 A72D52 16 24 a6a5a4a3a2a1 2 bhdech A64 A83 16 24 a52d4a32a12 4 eegd A54D4 A92D6 16 24 a5a5a42a3 4 dddf A54D4 A92D6 16 24 a52d4a32a12 8 eebd A54D4 D64 16 24 d44a18 48 bc D46 D64 16 24 E a46 240 a 2A46 16 25 a7a34a12 8 fff 16 25 a7a3a3a3a22a12 2 efggch 16 25 a5d43a13 36 bcb 16 25 d5a42a3a3a1a1 2 bbcbdb 16 25 d5d4a33a13a1 6 dgbec 16 25 d5a5a3a24a1 4 egcad 16 25 a6a5a4a2a2a1a1 1 bhdcagh 16 25 d5a4a34 4 bbb 16 25 d5a42a3a22a12 2 cbcae 16 25 a6d4a4a3a2a1a1 1 ahcgafe 16 25 a52a5a3a2 2 deeb 16 25 a6a5a3a3a2a2 1 bhfebc 16 25 a6a5a3a3a2a1a1a1 1 bhegchhh 16 25 a6a42a32a1 2 bcee 16 25 a5a5a5a22a12a1 2 defchd 16 25 a52d4a3a22 4 cfcb 16 25 a5a5d4a3a22 2 ccdga 16 25 a6d4a32a22a2 2 ahgab 16 25 a5a5a4a4a2a1 1 ddddcg 16 25 a5a5a4a4a2a1 1 ddadad 16 25 a6a4a4a3a2a2a1 1 bddecah 16 25 a5d4a42a3a1 2 edccb 16 25 a5d42a32a1a1a1 2 cebecf 16 25 a52a4a32a12 2 haeh 16 25 a52a4a3a3a12 2 ddebd 16 25 a5a5a4a3a3a1a1 1 eddefdg 16 25 a52a34 8 cf 16 25 a52a4a3a22a12 2 hdcch 16 25 d4a42a4a32 2 hcbb 17 24 a7a42a32 4 bfc A72D52 A83 17 24 a62a4a3a12 4 hebb A72D52 A83 17 24 a6a5a5a3a2 2 dddcg A64 A92D6 17 24 a62a32a22 4 hah A64 A92D6 17 24 a54 24 a A64 D64 17 25 a7a4a4a3a2a1 1 bfgcgb 17 25 a62a32a2a1 2 hcfa 17 25 a7a42a22a2a1 2 bfhea 17 25 d5d4a42a22 2 abbb 17 25 d5a5a4a3a3a1 1 bbbaab 17 25 d5a5a4a3a2a2a1 1 bbbaedb 17 25 a6a5d4a3a2a1 1 ecdafb 17 25 a6a5a42a12 2 ddeb 17 25 a6a5a4a3a3 1 dcgbc 17 25 a6a5a4a3a3 1 dcfcc 17 25 a6a5a4a3a3 1 dceac 17 25 a52d4a4a3 2 cdbb 17 25 a6a4a4a4a3a1 1 deffab 18 21 a53d4d4a1 12 bcab A54D4 18 22 a62a42a3 4 caa A64 18 23 a6d5a4a4a2a12 2 bhaaba A72D52 18 23 a62d4a4a12 4 ceba A72D52 18 23 d5a52d4a12a12 4 ebcfa A72D52 18 23 a7a5a42a12a1 4 dfcag A72D52 18 23 a7d42a32a12 8 cgfa A72D52 18 23 d52a34a12 16 gea A72D52 18 24 a7a52a22 8 ffa A83 A83 18 24 a7a5a42a1 4 dfcg A72D52 A92D6 18 24 a7a5d4a3a1a1a1 2 eggfdfc A72D52 A92D6 18 24 a6d5a4a4a2 2 bhaab A72D52 A92D6 18 24 d52a34 16 gb A72D52 D64 18 24 d5a52d4a12 4 ebbc A72D52 D64 18 24 a52a5d4a3a1 4 gbcdb A54D4 A11D7E6 18 24 a54a14 48 cb A54D4 E64 18 25 d6a34a3a12 8 ddcd 18 25 a7a52a14 8 fge 18 25 a7a5d4a22a1 2 egfbc 18 25 a7d42a3a14 4 cgef 18 25 a7a5a4a3a2 1 dfchb 18 25 a62a5a2a1a1 2 deaed 18 25 a7a5a4a3a1a1a1 1 dgchfeg 18 25 a6d5a4a3a2a1a1 1 bhaebfd 18 25 d5a5a5a4a1a1 1 dbcacc 18 25 a7a5a32a3a1 2 dggfd 18 25 a7a5a32a3a1 2 dfhfg 18 25 a6a6a4a3a2a1 1 cdchbg 18 25 a6a5a5a4a1 1 aeecc 18 25 d5a5d4a32a12a1 2 ebcefb 18 25 d5d42a32a3 4 cbbc 18 25 d5a5a42a3a1 2 dbadf 18 25 d5a5a42a3a1 2 dbabb 18 25 a6a5d4a4a2a1 1 cgeabf 18 25 a6a5a4a4a3 1 cfacg 18 25 a52d42a3a12 2 bgff 18 25 a6a42a42a1 2 bcag 19 24 a8a42a32 4 hhb A83 A92D6 19 24 a7a6a5a2a1 2 dfceb A83 A92D6 19 24 a6a6a5a4a1 2 eeaha A64 A11D7E6 19 24 E d46 2160 d 2D46 19 24 E a54d4 48 aa 2A54D4 19 25 d6a42a4a22 2 addc 19 25 a8a4a4a3a2a1 1 hghbfb 19 25 a8a42a22a2a1 2 hhgeb 19 25 a7a6a4a2a2a1 1 dfheeb 19 25 d52a4a4a22 2 bbec 19 25 a6d5d4a4a2 1 bcaec 19 25 a6d5a5a3a2a1 1 bdabca 19 25 a7a52a3a12 2 dcab 19 25 d5a53a1 3 aaa 19 25 a7a52a22a12 2 dcdb 19 25 a7a5a4a4a2 1 achgc 19 25 a7d4a4a4a3 1 ccefb 19 25 a62a5a3a2 2 fbad 19 25 a6a53 6 cb 19 25 a6a52a5 2 ecb 19 25 a6a5a5d4a2 1 babcc 19 25 d5a52a4a22 2 dadc 20 20 d5a54 16 ad A54D4 20 20 d5d45 120 dc D46 20 21 a63d4a2 6 aaa A64 20 22 a7d5a5a3a3a1 2 bgecad A72D52 20 22 d52a52a3 8 bba A72D52 20 22 a72a32a3a12 8 gdae A72D52 20 23 a7a62a12a12 4 ecga A83 20 23 a72a4a3a12 4 faea A83 20 23 a8a52a22a12 4 dhba A83 20 24 a72d4a14 8 gff A92D6 A92D6 20 24 d6a52a32 8 edd A92D6 A92D6 20 24 a8a52a3 4 dge A92D6 A92D6 20 24 d6a52a32 4 edc D64 A92D6 20 24 d6d43a16 12 bcb D64 D64 20 24 a7d5a5a3a1a1a1 2 cgefecd A72D52 A11D7E6 20 24 d5d5a52a12 4 bced A72D52 A11D7E6 20 24 a7d5d4a3a3 2 bgecf A72D52 A11D7E6 20 24 a6a6d5a4 2 aaeb A72D52 A11D7E6 20 24 d52a52a12 8 cbc A72D52 E64 20 24 a62a52 4 ch A64 A122 20 24 d46 48 c D46 D83 20 25 a8a52a12a12 2 dhfg 20 25 d6a52a3a12a1a1 2 eddfeb 20 25 d6a5d4a32a1 2 ccdcd 20 25 a7a7a3a22a12 2 fggbg 20 25 a8d4a4a3a3 1 chbff 20 25 d52a5d4a12a1 2 bbecb 20 25 a7d5a42a1a1 2 cfbcf 20 25 a7a6a5a3 1 ecfe 20 25 a62d5a3a12 2 aedd 20 25 a7d5a4a3a3 1 bfbfc 20 25 a7a6a5a2a1a1a1 1 echbgfe 20 25 a7a6a4a4a1 1 ecbad 20 25 a62a6a3a1 2 abee 20 25 a6a6a6a3a1 1 caceg 20 25 a6d5a5a4a2a1 1 aeebad 20 25 a6a6a5d4a1 1 abfhd 20 25 a7a52a4a12 2 ehag 20 25 d5a5a5a42 2 fbdb 21 24 a8a6a5a2a1 2 ehbga A83 A11D7E6 21 24 a72a42 4 cg A72D52 A122 21 25 d6a6a4a4a2 1 cdcdd 21 25 a8a6a4a3a1 1 eggbb 21 25 a7a6d5a2a1a1 1 aecfba 21 25 a6d5d5a4a2 1 baacc 21 25 a7d5a5a4a1 1 acbdb 21 25 a7a6a5a4 1 cgbe 22 21 a7d52d4a3 4 beac A72D52 22 22 a8a62a3 4 bba A83 22 23 a8a7a5a12a1 2 cgeac A92D6 22 23 a8a6d5a2a12 2 abhba A92D6 22 23 a7d6a5a3a12a1 2 dgdfab A92D6 22 23 a9a5a42a12 4 fgba A92D6 22 23 d6d5a52a12 4 bdcd D64 22 23 d54a12 48 bd D64 22 24 a9a5d4a3a1a1 2 gfffeb A92D6 A11D7E6 22 24 a7d6a5a3a1 2 dgcfe A92D6 A11D7E6 22 24 a8a6d5a2 2 abhb A92D6 A11D7E6 22 24 d6d5a52 4 bdc D64 A11D7E6 22 24 d54 48 b D64 E64 22 24 a8a7a4a3 2 chbg A83 A122 22 24 a7d52a32 4 eed A72D52 D83 22 24 a72d42 8 fd A72D52 D83 22 24 E a64 24 a 2A64 22 25 e6d42a33 12 cbc 22 25 e6a5a42a3a1 2 dbace 22 25 e6a52a24 8 fba 22 25 d7a36 24 ge 22 25 a9a5d4a22 2 gfgb 22 25 a9a5a4a3a1a1 1 ffbgbc 22 25 a9a5a32a3 2 fggf 22 25 a8a7a4a2a1 1 chbbc 22 25 d6d5a5a3a3a1 1 bdcdbe 22 25 a7d6a32a3a12 2 dgfeb 22 25 a8d5a5a3a1 1 agffe 22 25 a8d5a5a2a2a1 1 ahfbab 22 25 a8a6a5a3 1 bbgg 22 25 a8a6a5a3 1 cbef 22 25 d6a52d4a3 2 bcdb 22 25 d6a53a13 6 cdb 22 25 a8a52d4 2 bfe 22 25 a8a6a4a4a1 1 cbbbc 22 25 a7a7a5a3a1 1 ghefc 22 25 d53a33 6 ec 22 25 a7d5d42a3 2 bffc 23 24 a82a32 4 hb A92D6 A122 23 24 a9a6a5a2 2 cgbc A92D6 A122 23 24 a73 12 a A83 D83 23 25 d7a44 4 bd 23 25 a9a6a4a3 1 cfgb 23 25 a8d52a22 2 ebe 23 25 d6a6a6a4 1 accf 23 25 d6a62a4 2 bce 23 25 a8a7d4a3 1 fbbb 23 25 a72d5a3 2 baa 23 25 a62d52 2 cb 24 20 a72d5d5 4 cda A72D52 24 21 a82d4a4 4 baa A83 24 22 a9a7d4a3a1 2 fffad A92D6 24 22 a72d6a3 4 cfa A92D6 24 22 d62d42a3a12 4 cbdb D64 24 24 d7a52a5a1 4 cdea A11D7E6 A11D7E6 24 24 a9d52a12a1 4 efec A11D7E6 A11D7E6 24 24 a7e6d4a32 4 bgce A11D7E6 A11D7E6 24 24 e6a62a4 4 eaa A11D7E6 A11D7E6 24 24 e6d5a52a12 4 cbca E64 A11D7E6 24 24 d62d42a14 4 cbb D64 D83 24 24 a72d6a12 4 dfd A92D6 D83 24 24 a72d5d4 4 fde A72D52 A15D9 24 25 d7a52d4a1a1 2 bdeeb 24 25 a9a7a32a1 2 fgcd 24 25 d6d52a5a1 2 bcba 24 25 a7d6d5a3a1a1 1 cedeeb 24 25 a8a7a6a1 1 aebd 24 25 a7d6a5d4a1 1 cfdfb 24 25 a72a7a12 2 edd 24 25 a8a7d4a4 1 bfgb 24 25 a7a7d5a4 1 ccgb 24 25 a72d5a4 2 cea 25 24 a10a6a5a1 2 hgbb A11D7E6 A122 25 24 a8a72 4 eb A83 A15D9 25 24 E a72d52 8 aa 2A72D52 25 25 d7a6a6a2a2 1 adedc 25 25 a10a6a4a2a1 1 hgcea 25 25 e6a6d5a4a2 1 acaea 25 25 e6a62d4 2 bca 25 25 a7e6a5a4a1 1 acaeb 25 25 a10a5a4a4 1 gbcb 25 25 a8d6a6a2 1 dbfc 25 25 a9a6d5a2a1 1 bfbeb 25 25 a9a62a2 2 agc 25 25 a9d5a5a4 1 bbbe 26 19 a72d6d5 4 dae A72D52 26 21 a9d6a5d4 2 cfea A92D6 26 23 a9d6d5a12a1 2 cffab A11D7E6 26 23 a10a6d5a12 2 bbga A11D7E6 26 23 a8e6a6a2a12 2 ahaba A11D7E6 26 23 d7a7d5a3a12 2 edeea A11D7E6 26 23 e6d6a52a12 4 dbea A11D7E6 26 23 e6d53a12 12 bce E64 26 24 a92a22 8 ga A122 A122 26 24 d7a7d5a3 2 eddc A11D7E6 D83 26 24 a9d6a5a3 2 dfbd A92D6 A15D9 26 24 a9a8a5 2 ebc A92D6 A15D9 26 24 a72d52 8 ce A72D52 D10E72 26 25 d7d52a3a3 2 bdba 26 25 a10d5a5a2a1 1 bgbbb 26 25 a7d62a3 2 bcc 26 25 a9d6d4a3a1 1 cfbeb 26 25 a9a7a6 1 egb 26 25 a9a7a6 1 efb 26 25 a7d6d5a5a1 1 dffeb 27 24 a82a7 4 ga A83 A17E7 27 25 a8d7a4a4 1 dbde 27 25 a92a3a12 2 baa 27 25 a10a62a1 2 eca 28 20 a92d5a12 4 fac A92D6 28 20 d63d5a12 6 bdb D64 28 22 a9e6d5a3a1 2 cfead A11D7E6 28 22 a9d7a5a3 2 dfca A11D7E6 28 22 a11d5a5a3a1 2 fbeae A11D7E6 28 22 e62a52a3 8 bae E64 28 23 a10a9a2a12a1 2 bgaaf A122 28 23 a11a7a4a12 2 gcaa A122 28 24 d8d44 8 cb D83 D83 28 24 a9d7a5a1a1 2 efede A11D7E6 A15D9 28 24 a11d5d4a3 2 fbeb A11D7E6 A15D9 28 24 a9a7d6a1 2 fefc A92D6 A17E7 28 24 a9a7d6a1 2 fbfc A92D6 D10E72 28 24 d62d6d4a12 2 bbbb D64 D10E72 28 24 E a83 12 a 2A83 28 25 d8a52a5a1 2 ddbe 28 25 a11a7a22a12 2 gbaf 28 25 a11d5a4a3 1 fcbb 28 25 e6d6d5a5a1 1 cceab 28 25 d7d6a5a5 1 cdcb 28 25 a8a7e6a1a1 1 aeedc 28 25 a9e6a42a1 2 cgbc 28 25 a10a8a4a1 1 bbae 28 25 a92a4a12 2 gaf 28 25 a9a8d5a1 1 fbcd 29 24 a11a8a3 2 bca A122 A15D9 29 25 a10d6a6 1 fbe 30 21 d7a7e6d4 2 cada A11D7E6 30 22 a102a3 4 ba A122 30 23 d72a7a12 4 dcd D83 30 23 d8a72a12 4 ddd D83 30 24 d8a72 4 dd D83 A15D9 30 24 a11d6a5a1 2 fbba A11D7E6 A17E7 30 24 a10e6a6 2 agb A11D7E6 A17E7 30 24 d7a7e6a3 2 caed A11D7E6 D10E72 30 24 a9e6d6a1 2 efea A11D7E6 D10E72 30 24 e62d52 8 ca E64 D10E72 30 25 d8a7d5a3 1 cbdd 30 25 a10a9a4 1 bca 30 25 d7a7d52 2 cae 31 24 a12a7a4 2 gbf A122 A17E7 31 24 E d64 24 d 2D64 31 24 E a92d6 4 aa 2A92D6 31 25 a12a6a5 1 gba 31 25 a8e62a22 2 aaa 31 25 a10e6d5a2 1 ebba 31 25 a11a8d4 1 bea 31 25 a11a7d5 1 bba 31 25 a8a8d7 1 dcb 31 25 a82d7 2 ca 32 18 a92d7 4 da A92D6 32 18 d7d63 6 db D64 32 20 a11e6d5a3 2 ebaa A11D7E6 32 21 a12a8d4 2 bba A122 32 22 d8d62a3a12 2 bbdb D83 32 24 d8d62a12a12 2 bbba D83 D10E72 32 24 d64 8 b D64 D122 32 25 a9d8a5a1a1 1 dfecb 32 25 a12a7d4 1 bbf 32 25 a9d7d6a1 1 ceeb 34 19 a11d7d6 2 cea A11D7E6 34 19 e63d6 12 ae E64 34 23 a11d8a3a12 2 dbca A15D9 34 23 a13a8a12a1 2 caac A15D9 34 23 d9a72a12 4 eea A15D9 34 24 a13d6a3a1 2 bbcb A15D9 A17E7 34 24 d9a72 4 ed A15D9 D10E72 34 24 a11d7d5 2 eee A11D7E6 D122 34 25 e7a7d5a5 1 cbca 34 25 e7a72a3a1 2 dcab 34 25 d9a7d5a3 1 ddeb 34 25 a13a7a3a1 1 cfcc 34 25 d72e6a3 2 aca 34 25 d8a7e6a3 1 edda 34 25 a11d62 2 cb 35 24 a112 4 a A122 D122 35 25 a12d7a4 1 ebc 36 20 d82d5d4 2 bda D83 36 22 a13d7a3a1 2 ebab A15D9 36 24 a9e7a7 2 cfa D10E72 A17E7 36 24 e7d6d6d4a1 2 bbbaa D10E72 D10E72 36 24 d82d42 2 bb D83 D122 36 25 a12a10a1 1 aab 37 24 E e64 48 e 2E64 37 24 E a11d7e6 2 aaa 2A11D7E6 37 25 a13e6a4a1 1 baba 38 17 a11d8e6 2 dae A11D7E6 38 21 a11d9d4 2 dea A15D9 38 23 d9e62a12 4 add D10E72 38 23 a9e7e6a12 2 aecd D10E72 38 23 a14e6a2a12 2 bfaa A17E7 38 23 a11e7a5a12 2 cbba A17E7 38 24 a11d9a3 2 eeb A15D9 D122 38 24 a12a11 2 af A122 A24 38 25 d9d7a7 1 cdb 38 25 a11d8d5 1 dbc 40 20 a15d5d5 2 bba A15D9 40 22 d8e7d6a3a1 1 bbada D10E72 40 22 d10d62a3 2 bbd D10E72 40 22 a15d6a3 2 bca A17E7 40 24 d10d62a12 2 bba D10E72 D122 40 24 E a122 4 a 2A122 40 25 d10a9a5 1 dbb 40 25 a15d5a4 1 bca 40 25 e7e62a5 2 aaa 40 25 a9e7d7 1 aca 40 25 a11e7d5a1 1 aeba 40 25 a14a9 1 ac 41 24 a15a8 2 ac A15D9 A24 42 21 a13e7d4 2 aba A17E7 43 24 a16a7 2 fa A17E7 A24 43 24 E d83 6 d 2D83 44 16 d9d82 2 db D83 44 20 e72d6d5 2 bad D10E72 44 24 d82d8 2 ba D83 D16E8 44 24 d83 6 a D83 E83 46 23 d11a11a12 2 ebd D122 46 24 a15d8 2 cb A15D9 D16E8 46 25 d11a7e6 1 dab 47 25 a16d7 1 ca 48 18 d10e7d7a1 1 abda D10E72 48 18 a17d7a1 2 cac A17E7 48 22 d102a3a12 2 bdb D122 48 24 a15e7a1 2 bcc A17E7 D16E8 48 24 d10e7d6a1 1 abaa D10E72 D16E8 48 24 d8e72a12 2 aaa D10E72 D16E8 48 25 a13d10a1 1 bcb 49 24 E a15d9 2 aa 2A15D9 50 15 a15d10 2 ba A15D9 52 20 d12d8d5 1 bad D122 52 23 a19a4a12 2 caa A24 52 24 d12d8d4 1 bab D122 D16E8 55 24 E d10e72 2 dd 2D10E72 55 24 E a17e7 2 aa 2A17E7 55 25 a19d5 1 aa 56 14 d11e72 2 da D10E72 56 21 a20d4 2 aa A24 58 25 e8a11e6 1 aac 58 25 a11d13 1 bb 60 24 e8d82 2 aa E83 D16E8 62 23 a15e8a12 2 cbd D16E8 64 22 e8e72a3 2 aae E83 64 22 d14e7a3a1 1 abda D16E8 67 24 E d122 2 d 2D122 68 12 d13d12 1 db D122 68 20 d12e8d5 1 aad D16E8 68 24 d122 2 b D122 D24 71 24 a23 2 a A24 D24 76 16 E d16d9 1 bd D16E8 76 16 E d9e82 2 ea E83 76 24 d16d8 1 ba D16E8 D24 76 24 E a24 2 a 2A24 91 24 E e83 6 e 2E83 91 24 E d16e8 1 dd 2D16E8 92 8 E d17e8 1 da D16E8 100 20 d20d5 1 ad D24 139 24 E d24 1 d 2D24 140 0 E d25 1 d D24