Simon King
David J. Green
Cohomology
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Singular
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Cohomology of group number 1319 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t8 + t7 − 2·t6 + 2·t5 − 3·t4 − 3·t3 − t2 − 2·t − 1) |
| (t + 1)2 · (t − 1)4 · (t2 + 1)3 |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 14 minimal generators of maximal degree 5:
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- a_3_5, a nilpotent element of degree 3
- a_3_6, a nilpotent element of degree 3
- a_3_9, a nilpotent element of degree 3
- b_3_7, an element of degree 3
- b_3_8, an element of degree 3
- b_3_10, an element of degree 3
- c_4_17, a Duflot regular element of degree 4
- c_4_18, a Duflot regular element of degree 4
- c_4_19, a Duflot regular element of degree 4
- b_5_29, an element of degree 5
Ring relations
There are 41 minimal relations of maximal degree 10:
- a_1_22 + a_1_1·a_1_2 + a_1_12
- a_1_3·b_1_0 + a_1_1·b_1_0 + a_1_12
- a_1_2·b_1_0 + a_1_32 + a_1_12
- a_1_13
- a_1_1·a_1_32
- a_1_2·a_1_32 + a_1_12·a_1_3 + a_1_12·a_1_2
- b_1_0·a_3_6 + b_1_0·a_3_5 + a_1_1·b_1_03 + a_1_3·a_3_9 + a_1_3·a_3_6 + a_1_3·a_3_5
+ a_1_2·a_3_6 + a_1_2·a_3_5 + a_1_1·a_3_9 + a_1_1·a_3_5
- a_1_3·b_3_7 + a_1_1·b_3_7 + a_1_2·a_3_6 + a_1_2·a_3_5 + a_1_1·a_3_6 + a_1_1·a_3_5
- b_1_0·a_3_6 + b_1_0·a_3_5 + a_1_2·b_3_7 + a_1_1·b_1_03
- b_1_0·a_3_9 + b_1_0·a_3_6 + b_1_0·a_3_5 + a_1_1·b_3_8 + a_1_1·b_1_03 + a_1_3·a_3_6
+ a_1_3·a_3_5 + a_1_2·a_3_9 + a_1_2·a_3_6 + a_1_2·a_3_5 + a_1_1·a_3_6 + a_1_1·a_3_5
- b_1_0·a_3_9 + b_1_0·a_3_6 + b_1_0·a_3_5 + a_1_3·b_3_8 + a_1_1·b_1_03 + a_1_3·a_3_6
+ a_1_3·a_3_5 + a_1_2·a_3_9 + a_1_2·a_3_6 + a_1_1·a_3_9 + a_1_1·a_3_6 + a_1_1·a_3_5
- a_1_2·b_3_8 + a_1_3·a_3_6 + a_1_2·a_3_9 + a_1_2·a_3_6 + a_1_2·a_3_5 + a_1_1·a_3_9
+ a_1_1·a_3_6
- b_1_0·a_3_5 + a_1_1·b_3_10 + a_1_1·b_3_7 + a_1_1·a_3_9 + a_1_1·a_3_6
- b_1_0·a_3_6 + a_1_3·b_3_10 + a_1_1·b_3_7 + a_1_1·b_1_03 + a_1_3·a_3_5 + a_1_1·a_3_9
+ a_1_1·a_3_6 + a_1_1·a_3_5
- a_1_2·b_3_10 + a_1_3·a_3_6 + a_1_3·a_3_5 + a_1_2·a_3_5 + a_1_1·a_3_6 + a_1_1·a_3_5
- a_1_2·a_1_3·a_3_6 + a_1_2·a_1_3·a_3_5 + a_1_1·a_1_3·a_3_6 + a_1_1·a_1_2·a_3_6
+ a_1_1·a_1_2·a_3_5 + a_1_12·a_3_6
- a_1_2·a_1_3·a_3_6 + a_1_2·a_1_3·a_3_5 + a_1_1·a_1_3·a_3_9 + a_1_1·a_1_3·a_3_5
+ a_1_12·a_3_9 + a_1_12·a_3_6 + a_1_12·a_3_5
- a_1_2·a_1_3·a_3_9 + a_1_1·a_1_3·a_3_6 + a_1_1·a_1_2·a_3_9 + a_1_1·a_1_2·a_3_5
+ a_1_12·a_3_9 + a_1_12·a_3_6
- a_3_9·b_3_10 + a_3_9·b_3_8 + a_3_9·b_3_7 + a_3_6·b_3_10 + a_3_6·b_3_7 + a_3_5·b_3_8
+ a_1_1·b_1_02·b_3_10 + a_1_1·b_1_02·b_3_8 + a_3_92 + a_3_62 + a_3_5·a_3_9 + c_4_18·a_1_1·b_1_0 + c_4_17·a_1_1·b_1_0 + c_4_17·a_1_2·a_1_3
- b_3_102 + b_3_82 + b_3_72 + b_1_03·b_3_8 + b_1_03·b_3_7 + a_1_1·b_1_02·b_3_7
+ a_1_1·b_1_05 + a_3_92 + a_3_62 + a_3_52 + c_4_18·b_1_02 + c_4_17·b_1_02 + c_4_17·a_1_32 + c_4_17·a_1_1·a_1_2 + c_4_17·a_1_12
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_1_1·b_1_02·b_3_10
+ a_1_1·b_1_02·b_3_8 + a_1_1·b_1_02·b_3_7 + a_1_1·b_1_05 + c_4_17·b_1_02 + c_4_18·a_1_32 + c_4_18·a_1_12 + c_4_17·a_1_32 + c_4_17·a_1_1·a_1_2
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_9·b_3_10 + a_3_9·b_3_7
+ a_3_6·b_3_10 + a_3_5·b_3_10 + a_3_5·b_3_8 + a_1_1·b_1_02·b_3_8 + a_1_1·b_1_02·b_3_7 + a_1_1·b_1_05 + a_3_92 + a_3_6·a_3_9 + a_3_62 + a_3_5·a_3_6 + a_3_52 + a_1_12·a_1_2·a_3_5 + c_4_17·b_1_02 + c_4_18·a_1_2·a_1_3 + c_4_17·a_1_2·a_1_3 + c_4_17·a_1_12
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_9·b_3_10 + a_3_9·b_3_7
+ a_3_5·b_3_8 + a_1_1·b_1_02·b_3_10 + a_1_1·b_1_02·b_3_8 + a_1_1·b_1_02·b_3_7 + a_1_1·b_1_05 + a_3_92 + a_3_6·a_3_9 + a_3_62 + a_3_52 + a_1_12·a_1_2·a_3_5 + c_4_17·b_1_02 + c_4_18·a_1_1·a_1_2 + c_4_18·a_1_12 + c_4_17·a_1_32 + c_4_17·a_1_2·a_1_3
- a_3_52 + c_4_19·a_1_12 + c_4_17·a_1_32
- a_3_5·b_3_10 + a_3_5·b_3_7 + a_1_1·b_1_02·b_3_10 + a_3_5·a_3_9 + a_3_5·a_3_6
+ a_1_12·a_1_2·a_3_5 + c_4_19·a_1_1·b_1_0 + c_4_17·a_1_1·b_1_0 + c_4_17·a_1_1·a_1_3 + c_4_17·a_1_12
- b_3_102 + b_3_72 + b_1_03·b_3_10 + a_1_1·b_1_02·b_3_10 + a_1_1·b_1_02·b_3_8
+ a_3_92 + a_3_62 + a_1_12·a_1_2·a_3_5 + c_4_19·b_1_02 + c_4_17·b_1_02 + c_4_17·a_1_12
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_9·b_3_10 + a_3_9·b_3_7
+ a_3_6·b_3_10 + a_3_6·b_3_8 + a_3_6·b_3_7 + a_3_5·b_3_10 + a_3_5·b_3_7 + a_1_1·b_1_05 + a_3_62 + a_3_52 + c_4_17·b_1_02 + c_4_19·a_1_1·a_1_3 + c_4_17·a_1_32 + c_4_17·a_1_2·a_1_3
- a_3_9·b_3_10 + a_3_9·b_3_7 + a_3_5·b_3_8 + a_3_92 + a_3_6·a_3_9 + a_3_52
+ a_1_12·a_1_2·a_3_5 + c_4_19·a_1_32 + c_4_17·a_1_2·a_1_3 + c_4_17·a_1_1·a_1_2 + c_4_17·a_1_12
- a_3_6·b_3_8 + a_3_5·b_3_8 + a_1_1·b_1_02·b_3_8 + a_3_6·a_3_9 + a_3_62 + a_3_5·a_3_6
+ c_4_19·a_1_2·a_1_3 + c_4_18·a_1_1·a_1_3 + c_4_18·a_1_12 + c_4_17·a_1_2·a_1_3 + c_4_17·a_1_1·a_1_3 + c_4_17·a_1_12
- a_3_9·b_3_10 + a_3_9·b_3_7 + a_3_5·b_3_8 + a_3_6·a_3_9 + c_4_19·a_1_1·a_1_2
+ c_4_18·a_1_12 + c_4_17·a_1_2·a_1_3 + c_4_17·a_1_12
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_9·b_3_10 + a_3_9·b_3_7
+ a_3_6·b_3_10 + a_3_6·b_3_8 + a_3_6·b_3_7 + a_3_5·b_3_8 + a_1_1·b_5_29 + a_1_1·b_1_02·b_3_10 + a_1_1·b_1_02·b_3_7 + a_1_1·b_1_05 + a_3_62 + a_3_5·a_3_9 + a_3_5·a_3_6 + a_3_52 + a_1_12·a_1_2·a_3_5 + c_4_17·b_1_02 + c_4_18·a_1_1·a_1_3 + c_4_18·a_1_12 + c_4_17·a_1_32 + c_4_17·a_1_1·a_1_3 + c_4_17·a_1_1·a_1_2
- b_3_102 + b_3_8·b_3_10 + b_3_7·b_3_8 + b_3_72 + b_1_0·b_5_29 + b_1_03·b_3_10
+ b_1_03·b_3_7 + a_3_9·b_3_8 + a_3_6·b_3_8 + a_3_5·b_3_10 + a_3_5·b_3_7 + a_1_1·b_1_02·b_3_8 + a_1_1·b_1_05 + a_3_92 + a_3_62 + a_3_5·a_3_9 + a_3_5·a_3_6 + a_3_52 + a_1_12·a_1_2·a_3_5 + c_4_18·a_1_12 + c_4_17·a_1_32 + c_4_17·a_1_1·a_1_3 + c_4_17·a_1_1·a_1_2
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_5·b_3_10 + a_3_5·b_3_8
+ a_3_5·b_3_7 + a_1_3·b_5_29 + a_1_1·b_1_02·b_3_8 + a_1_1·b_1_05 + a_3_92 + a_3_5·a_3_6 + c_4_17·b_1_02 + c_4_18·a_1_1·a_1_3 + c_4_17·a_1_2·a_1_3
- b_3_102 + b_3_82 + b_1_03·b_3_10 + b_1_03·b_3_7 + a_3_9·b_3_10 + a_3_9·b_3_7
+ a_3_6·b_3_10 + a_3_5·b_3_10 + a_3_5·b_3_8 + a_1_2·b_5_29 + a_1_1·b_1_02·b_3_8 + a_1_1·b_1_02·b_3_7 + a_1_1·b_1_05 + a_3_62 + a_3_5·a_3_6 + a_1_12·a_1_2·a_3_5 + c_4_17·b_1_02 + c_4_18·a_1_1·a_1_3 + c_4_17·a_1_32 + c_4_17·a_1_1·a_1_3
- a_1_2·a_3_6·a_3_9 + a_1_2·a_3_5·a_3_9 + a_1_1·a_3_6·a_3_9 + a_1_1·a_3_5·a_3_6
+ c_4_19·a_1_1·a_1_2·a_1_3 + c_4_19·a_1_12·a_1_3 + c_4_18·a_1_1·a_1_2·a_1_3 + c_4_18·a_1_12·a_1_3 + c_4_18·a_1_12·a_1_2 + c_4_17·a_1_1·a_1_2·a_1_3
- a_3_9·b_5_29 + a_1_1·b_1_0·b_3_7·b_3_8 + a_1_1·b_1_02·b_5_29 + a_1_1·b_1_04·b_3_10
+ a_1_1·b_1_04·b_3_7 + a_1_12·a_3_5·a_3_6 + c_4_19·a_1_1·b_3_10 + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_1·b_3_7 + c_4_18·a_1_1·b_3_10 + c_4_18·a_1_1·b_3_7 + c_4_17·a_1_1·b_3_8 + c_4_19·a_1_3·a_3_6 + c_4_19·a_1_3·a_3_5 + c_4_19·a_1_2·a_3_6 + c_4_19·a_1_2·a_3_5 + c_4_19·a_1_1·a_3_9 + c_4_19·a_1_1·a_3_6 + c_4_19·a_1_1·a_3_5 + c_4_18·a_1_3·a_3_9 + c_4_18·a_1_1·a_3_6 + c_4_17·a_1_3·a_3_9 + c_4_17·a_1_3·a_3_6 + c_4_17·a_1_3·a_3_5 + c_4_17·a_1_2·a_3_6 + c_4_17·a_1_2·a_3_5 + c_4_17·a_1_1·a_3_9 + c_4_17·a_1_1·a_3_5
- a_3_6·b_5_29 + a_1_1·b_1_0·b_3_7·b_3_10 + a_1_1·b_1_0·b_3_7·b_3_8
+ a_1_1·b_1_04·b_3_10 + a_1_1·b_1_04·b_3_8 + a_1_1·b_1_04·b_3_7 + a_1_12·a_3_5·a_3_9 + a_1_12·a_3_5·a_3_6 + c_4_19·a_1_1·b_3_10 + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_1·b_3_7 + c_4_19·a_1_1·b_1_03 + c_4_18·a_1_1·b_1_03 + c_4_17·a_1_1·b_3_10 + c_4_17·a_1_1·b_3_8 + c_4_17·a_1_1·b_3_7 + c_4_17·a_1_1·b_1_03 + c_4_19·a_1_3·a_3_6 + c_4_19·a_1_3·a_3_5 + c_4_19·a_1_2·a_3_5 + c_4_19·a_1_1·a_3_5 + c_4_18·a_1_3·a_3_9 + c_4_18·a_1_3·a_3_6 + c_4_18·a_1_2·a_3_9 + c_4_18·a_1_1·a_3_9 + c_4_18·a_1_1·a_3_5 + c_4_17·a_1_3·a_3_9 + c_4_17·a_1_3·a_3_5 + c_4_17·a_1_2·a_3_5 + c_4_17·a_1_1·a_3_9
- a_3_9·b_5_29 + a_3_5·b_5_29 + a_1_1·b_1_0·b_3_7·b_3_10 + a_1_1·b_1_04·b_3_8
+ a_1_12·a_3_5·a_3_6 + c_4_19·a_1_1·b_1_03 + c_4_18·a_1_1·b_3_10 + c_4_18·a_1_1·b_3_7 + c_4_18·a_1_1·b_1_03 + c_4_17·a_1_1·b_3_10 + c_4_17·a_1_1·b_3_7 + c_4_17·a_1_1·b_1_03 + c_4_19·a_1_3·a_3_6 + c_4_19·a_1_2·a_3_6 + c_4_19·a_1_2·a_3_5 + c_4_19·a_1_1·a_3_5 + c_4_18·a_1_3·a_3_9 + c_4_18·a_1_3·a_3_5 + c_4_18·a_1_1·a_3_6 + c_4_18·a_1_1·a_3_5 + c_4_17·a_1_3·a_3_6 + c_4_17·a_1_3·a_3_5 + c_4_17·a_1_2·a_3_6 + c_4_17·a_1_2·a_3_5 + c_4_17·a_1_1·a_3_9 + c_4_17·a_1_1·a_3_6
- b_3_10·b_5_29 + b_3_7·b_5_29 + b_1_02·b_3_7·b_3_10 + b_1_02·b_3_7·b_3_8
+ b_1_03·b_5_29 + b_1_05·b_3_10 + b_1_05·b_3_8 + b_1_05·b_3_7 + a_3_9·b_5_29 + a_1_1·b_1_0·b_3_7·b_3_8 + a_1_1·b_1_04·b_3_7 + a_1_1·b_1_07 + a_1_12·a_3_5·a_3_9 + c_4_19·b_1_0·b_3_10 + c_4_19·b_1_0·b_3_8 + c_4_19·b_1_0·b_3_7 + c_4_19·b_1_04 + c_4_18·b_1_04 + c_4_17·b_1_0·b_3_10 + c_4_17·b_1_0·b_3_8 + c_4_17·b_1_0·b_3_7 + c_4_17·b_1_04 + c_4_19·a_1_1·b_3_10 + c_4_19·a_1_1·b_3_7 + c_4_19·a_1_1·b_1_03 + c_4_17·a_1_1·b_3_10 + c_4_17·a_1_1·b_3_7 + c_4_17·a_1_1·b_1_03 + c_4_19·a_1_2·a_3_9 + c_4_19·a_1_2·a_3_6 + c_4_19·a_1_1·a_3_6 + c_4_19·a_1_1·a_3_5 + c_4_18·a_1_3·a_3_9 + c_4_18·a_1_3·a_3_6 + c_4_18·a_1_2·a_3_9 + c_4_18·a_1_1·a_3_9 + c_4_17·a_1_3·a_3_9 + c_4_17·a_1_3·a_3_6 + c_4_17·a_1_2·a_3_9 + c_4_17·a_1_2·a_3_6 + c_4_17·a_1_1·a_3_6 + c_4_17·a_1_1·a_3_5
- b_3_8·b_5_29 + b_1_02·b_3_7·b_3_8 + b_1_03·b_5_29 + b_1_05·b_3_10 + b_1_05·b_3_7
+ a_1_1·b_1_04·b_3_10 + a_1_1·b_1_07 + a_1_12·a_3_5·a_3_9 + a_1_12·a_3_5·a_3_6 + c_4_19·b_1_0·b_3_10 + c_4_19·b_1_0·b_3_8 + c_4_19·b_1_0·b_3_7 + c_4_18·b_1_0·b_3_10 + c_4_18·b_1_0·b_3_7 + c_4_17·b_1_0·b_3_8 + c_4_19·a_1_1·b_3_10 + c_4_19·a_1_1·b_3_7 + c_4_18·a_1_1·b_3_10 + c_4_18·a_1_1·b_3_8 + c_4_18·a_1_1·b_3_7 + c_4_18·a_1_1·b_1_03 + c_4_17·a_1_1·b_3_8 + c_4_17·a_1_1·b_1_03 + c_4_19·a_1_3·a_3_6 + c_4_19·a_1_3·a_3_5 + c_4_19·a_1_2·a_3_9 + c_4_19·a_1_2·a_3_6 + c_4_18·a_1_3·a_3_6 + c_4_18·a_1_3·a_3_5 + c_4_18·a_1_2·a_3_9 + c_4_18·a_1_2·a_3_6 + c_4_18·a_1_1·a_3_5 + c_4_17·a_1_2·a_3_9 + c_4_17·a_1_1·a_3_5
- b_5_292 + b_1_04·b_3_7·b_3_10 + b_1_04·b_3_7·b_3_8 + b_1_05·b_5_29 + b_1_07·b_3_10
+ b_1_07·b_3_7 + a_1_1·b_1_09 + c_4_19·b_1_03·b_3_8 + c_4_19·b_1_03·b_3_7 + c_4_18·b_1_03·b_3_10 + c_4_17·b_1_03·b_3_10 + c_4_17·b_1_03·b_3_8 + c_4_17·b_1_03·b_3_7 + c_4_19·a_1_1·b_1_02·b_3_7 + c_4_18·a_1_1·b_1_02·b_3_10 + c_4_18·a_1_1·b_1_02·b_3_8 + c_4_17·a_1_1·b_1_02·b_3_10 + c_4_17·a_1_1·b_1_02·b_3_8 + c_4_17·a_1_1·b_1_02·b_3_7 + c_4_18·a_1_12·a_1_2·a_3_5 + c_4_18·c_4_19·b_1_02 + c_4_17·c_4_19·b_1_02 + c_4_17·c_4_18·b_1_02 + c_4_172·b_1_02 + c_4_192·a_1_32 + c_4_192·a_1_12 + c_4_182·a_1_32 + c_4_182·a_1_12 + c_4_17·c_4_19·a_1_32 + c_4_17·c_4_19·a_1_1·a_1_2 + c_4_17·c_4_19·a_1_12 + c_4_17·c_4_18·a_1_12 + c_4_172·a_1_32 + c_4_172·a_1_1·a_1_2
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- The completion test was perfect: It applied in the last degree in which a generator or relation was found.
- The following is a filter regular homogeneous system of parameters:
- c_4_17, a Duflot regular element of degree 4
- c_4_18, a Duflot regular element of degree 4
- c_4_19, a Duflot regular element of degree 4
- b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 10].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- a_3_9 → 0, an element of degree 3
- b_3_7 → 0, an element of degree 3
- b_3_8 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- c_4_17 → c_1_04, an element of degree 4
- c_4_18 → c_1_24, an element of degree 4
- c_4_19 → c_1_24 + c_1_14, an element of degree 4
- b_5_29 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → c_1_3, an element of degree 1
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- a_3_9 → 0, an element of degree 3
- b_3_7 → c_1_22·c_1_3, an element of degree 3
- b_3_8 → c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_10 → c_1_2·c_1_32 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_17 → c_1_2·c_1_33 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_04, an element of degree 4
- c_4_18 → c_1_2·c_1_33 + c_1_24 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
- c_4_19 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
- b_5_29 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_1·c_1_2·c_1_33
+ c_1_1·c_1_22·c_1_32 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_34 + c_1_0·c_1_1·c_1_33 + c_1_0·c_1_12·c_1_32 + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·c_1_3 + c_1_04·c_1_3, an element of degree 5
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