# r2_tori.py: spans of products of isotypic projections; prime of the second engine p = 33554393
## 1. The representation of Remark 4.12(ii), exact arithmetic
EXACT  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxz   : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 1331 (non-zero 1128); dim over Q of their span = 142 of 144; kernel of Phi on the coefficient space has dimension 2   [0.2s]
   basis of the annihilator (functionals phi on End(H); the basis of H is indexed by the weights
   (tau_1, ..., tau_r); phi = sum of c * [entry (row, column)]):
     phi = -1*[(1, 0, -1),(-1, 0, 1)] +1*[(-1, 0, 1),(1, 0, -1)]
     phi = -1*[(1, 0, -1),(1, 0, -1)] +1*[(-1, 0, 1),(-1, 0, 1)]
   checked exactly: every functional of this basis vanishes on all 1128 non-zero products
EXACT  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxzx  : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 14641 (non-zero 11264); dim over Q of their span = 144 of 144; kernel of Phi on the coefficient space has dimension 0   [1.2s]
EXACT  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori xzx   : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 1331 (non-zero 1128); dim over Q of their span = 142 of 144; kernel of Phi on the coefficient space has dimension 2   [1.2s]
EXACT  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zx    : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 121 (non-zero 112); dim over Q of their span = 112 of 144; kernel of Phi on the coefficient space has dimension 32   [0.0s]
## 2. The group of order two in place of the circle, exact arithmetic
EXACT  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxz   : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 1331 (non-zero 1128); dim over Q of their span = 142 of 144; kernel of Phi on the coefficient space has dimension 2   [0.2s]
   basis of the annihilator (functionals phi on End(H); the basis of H is indexed by the weights
   (tau_1, ..., tau_r); phi = sum of c * [entry (row, column)]):
     phi = -1*[(1, 0, -1),(-1, 0, 1)] +1*[(-1, 0, 1),(1, 0, -1)]
     phi = -1*[(1, 0, -1),(1, 0, -1)] +1*[(-1, 0, 1),(-1, 0, 1)]
   checked exactly: every functional of this basis vanishes on all 1128 non-zero products
EXACT  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxzx  : 11 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]; products 14641 (non-zero 11264); dim over Q of their span = 144 of 144; kernel of Phi on the coefficient space has dimension 0   [1.2s]
## 3. Controls, exact arithmetic: classical representations (no factor z) and a representation without
##    coincidences of group elements
EXACT  V_1                                          dim   2  group Z*Z   tori zxz   : 2 isotypic components of ranks [1, 1]; products 8 (non-zero 8); dim over Q of their span = 4 of 4; kernel of Phi on the coefficient space has dimension 0   [0.0s]
EXACT  V_2                                          dim   3  group Z*Z   tori zxz   : 3 isotypic components of ranks [1, 1, 1]; products 27 (non-zero 22); dim over Q of their span = 9 of 9; kernel of Phi on the coefficient space has dimension 0   [0.0s]
EXACT  V_3                                          dim   4  group Z*Z   tori zxz   : 4 isotypic components of ranks [1, 1, 1, 1]; products 64 (non-zero 64); dim over Q of their span = 16 of 16; kernel of Phi on the coefficient space has dimension 0   [0.0s]
EXACT  V_2                                          dim   3  group Z*Z   tori zx    : 3 isotypic components of ranks [1, 1, 1]; products 9 (non-zero 8); dim over Q of their span = 8 of 9; kernel of Phi on the coefficient space has dimension 1   [0.0s]
EXACT  V_1 (x) z^1 (x) V_1 (x) z^-1 (x) V_1         dim   8  group Z*Z   tori zxz   : 8 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1]; products 512 (non-zero 512); dim over Q of their span = 64 of 64; kernel of Phi on the coefficient space has dimension 0   [0.0s]
EXACT  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1          dim  12  group Z*Z   tori zxz   : 12 isotypic components of ranks [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; products 1728 (non-zero 1408); dim over Q of their span = 144 of 144; kernel of Phi on the coefficient space has dimension 0   [0.2s]
## 4. Second engine (modulo p): the same cases again, then larger representations
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxz   : 11 isotypic components; rank modulo p of the span of the products = 142 of 144 (deficit 2)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxzx  : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxzxz : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  12  group Z*Z   tori zxzxzx: 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxz   : 11 isotypic components; rank modulo p of the span of the products = 142 of 144 (deficit 2)   [0.0s]
MOD p  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxzx  : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxzxz : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) t^1 (x) V_2 (x) t^1 (x) V_1          dim  12  group Z_2*Z tori zxzxzx: 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_1 (x) z^-1 (x) V_1         dim   8  group Z*Z   tori zxz   : 8 isotypic components; rank modulo p of the span of the products = 64 of 64 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_1 (x) z^-1 (x) V_1         dim   8  group Z*Z   tori zxzx  : 8 isotypic components; rank modulo p of the span of the products = 64 of 64 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_1 (x) z^-1 (x) V_1         dim   8  group Z*Z   tori zxzxz : 8 isotypic components; rank modulo p of the span of the products = 64 of 64 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_1 (x) z^-1 (x) V_1         dim   8  group Z*Z   tori zxzxzx: 8 isotypic components; rank modulo p of the span of the products = 64 of 64 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1          dim  12  group Z*Z   tori zxz   : 12 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1          dim  12  group Z*Z   tori zxzx  : 12 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1          dim  12  group Z*Z   tori zxzxz : 12 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1          dim  12  group Z*Z   tori zxzxzx: 12 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1         dim  12  group Z*Z   tori zxz   : 11 isotypic components; rank modulo p of the span of the products = 142 of 144 (deficit 2)   [0.0s]
MOD p  V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1         dim  12  group Z*Z   tori zxzx  : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1         dim  12  group Z*Z   tori zxzxz : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1         dim  12  group Z*Z   tori zxzxzx: 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 dim  12  group Z*Z   tori zxz   : 11 isotypic components; rank modulo p of the span of the products = 142 of 144 (deficit 2)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 dim  12  group Z*Z   tori zxzx  : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 dim  12  group Z*Z   tori zxzxz : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 dim  12  group Z*Z   tori zxzxzx: 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  z^1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^-1 dim  12  group Z*Z   tori zxz   : 11 isotypic components; rank modulo p of the span of the products = 142 of 144 (deficit 2)   [0.0s]
MOD p  z^1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^-1 dim  12  group Z*Z   tori zxzx  : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  z^1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^-1 dim  12  group Z*Z   tori zxzxz : 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  z^1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^-1 dim  12  group Z*Z   tori zxzxzx: 11 isotypic components; rank modulo p of the span of the products = 144 of 144 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  18  group Z*Z   tori zxz   : 16 isotypic components; rank modulo p of the span of the products = 320 of 324 (deficit 4)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  18  group Z*Z   tori zxzx  : 16 isotypic components; rank modulo p of the span of the products = 324 of 324 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  18  group Z*Z   tori zxzxz : 16 isotypic components; rank modulo p of the span of the products = 324 of 324 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1         dim  18  group Z*Z   tori zxzxzx: 16 isotypic components; rank modulo p of the span of the products = 324 of 324 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2         dim  27  group Z*Z   tori zxz   : 23 isotypic components; rank modulo p of the span of the products = 708 of 729 (deficit 21)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2         dim  27  group Z*Z   tori zxzx  : 23 isotypic components; rank modulo p of the span of the products = 729 of 729 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2         dim  27  group Z*Z   tori zxzxz : 23 isotypic components; rank modulo p of the span of the products = 729 of 729 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2         dim  27  group Z*Z   tori zxzxzx: 23 isotypic components; rank modulo p of the span of the products = 729 of 729 (deficit 0)   [0.1s]
MOD p  V_1 (x) z^1 (x) V_3 (x) z^-1 (x) V_1         dim  16  group Z*Z   tori zxz   : 16 isotypic components; rank modulo p of the span of the products = 256 of 256 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_3 (x) z^-1 (x) V_1         dim  16  group Z*Z   tori zxzx  : 16 isotypic components; rank modulo p of the span of the products = 256 of 256 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_3 (x) z^-1 (x) V_1         dim  16  group Z*Z   tori zxzxz : 16 isotypic components; rank modulo p of the span of the products = 256 of 256 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_3 (x) z^-1 (x) V_1         dim  16  group Z*Z   tori zxzxzx: 16 isotypic components; rank modulo p of the span of the products = 256 of 256 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_4 (x) z^-1 (x) V_1         dim  20  group Z*Z   tori zxz   : 19 isotypic components; rank modulo p of the span of the products = 398 of 400 (deficit 2)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_4 (x) z^-1 (x) V_1         dim  20  group Z*Z   tori zxzx  : 19 isotypic components; rank modulo p of the span of the products = 400 of 400 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_4 (x) z^-1 (x) V_1         dim  20  group Z*Z   tori zxzxz : 19 isotypic components; rank modulo p of the span of the products = 400 of 400 (deficit 0)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_4 (x) z^-1 (x) V_1         dim  20  group Z*Z   tori zxzxzx: 19 isotypic components; rank modulo p of the span of the products = 400 of 400 (deficit 0)   [0.0s]
MOD p  V_2 (x) z^1 (x) V_4 (x) z^-1 (x) V_2         dim  45  group Z*Z   tori zxz   : 41 isotypic components; rank modulo p of the span of the products = 2000 of 2025 (deficit 25)   [0.1s]
MOD p  V_2 (x) z^1 (x) V_4 (x) z^-1 (x) V_2         dim  45  group Z*Z   tori zxzx  : 41 isotypic components; rank modulo p of the span of the products = 2025 of 2025 (deficit 0)   [0.1s]
MOD p  V_2 (x) z^1 (x) V_4 (x) z^-1 (x) V_2         dim  45  group Z*Z   tori zxzxz : 41 isotypic components; rank modulo p of the span of the products = 2025 of 2025 (deficit 0)   [0.2s]
MOD p  V_2 (x) z^1 (x) V_4 (x) z^-1 (x) V_2         dim  45  group Z*Z   tori zxzxzx: 41 isotypic components; rank modulo p of the span of the products = 2025 of 2025 (deficit 0)   [0.3s]
MOD p  V_3 (x) z^1 (x) V_2 (x) z^-1 (x) V_3         dim  48  group Z*Z   tori zxz   : 39 isotypic components; rank modulo p of the span of the products = 2254 of 2304 (deficit 50)   [0.1s]
MOD p  V_3 (x) z^1 (x) V_2 (x) z^-1 (x) V_3         dim  48  group Z*Z   tori zxzx  : 39 isotypic components; rank modulo p of the span of the products = 2304 of 2304 (deficit 0)   [0.2s]
MOD p  V_3 (x) z^1 (x) V_2 (x) z^-1 (x) V_3         dim  48  group Z*Z   tori zxzxz : 39 isotypic components; rank modulo p of the span of the products = 2304 of 2304 (deficit 0)   [0.3s]
MOD p  V_3 (x) z^1 (x) V_2 (x) z^-1 (x) V_3         dim  48  group Z*Z   tori zxzxzx: 39 isotypic components; rank modulo p of the span of the products = 2304 of 2304 (deficit 0)   [0.4s]
MOD p  V_4 (x) z^1 (x) V_4 (x) z^-1 (x) V_4         dim 125  group Z*Z   tori zxz   : 109 isotypic components; rank modulo p of the span of the products = 15473 of 15625 (deficit 152)   [1.4s]
MOD p  V_4 (x) z^1 (x) V_4 (x) z^-1 (x) V_4         dim 125  group Z*Z   tori zxzx  : 109 isotypic components; rank modulo p of the span of the products = 15625 of 15625 (deficit 0)   [5.4s]
MOD p  V_4 (x) z^1 (x) V_4 (x) z^-1 (x) V_4         dim 125  group Z*Z   tori zxzxz : 109 isotypic components; rank modulo p of the span of the products = 15625 of 15625 (deficit 0)   [5.5s]
MOD p  V_4 (x) z^1 (x) V_4 (x) z^-1 (x) V_4         dim 125  group Z*Z   tori zxzxzx: 109 isotypic components; rank modulo p of the span of the products = 15625 of 15625 (deficit 0)   [9.5s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_1 dim  36  group Z*Z   tori zxz   : 28 isotypic components; rank modulo p of the span of the products = 1248 of 1296 (deficit 48)   [0.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_1 dim  36  group Z*Z   tori zxzx  : 28 isotypic components; rank modulo p of the span of the products = 1296 of 1296 (deficit 0)   [0.1s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_1 dim  36  group Z*Z   tori zxzxz : 28 isotypic components; rank modulo p of the span of the products = 1296 of 1296 (deficit 0)   [0.1s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_1 dim  36  group Z*Z   tori zxzxzx: 28 isotypic components; rank modulo p of the span of the products = 1296 of 1296 (deficit 0)   [0.2s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 dim  81  group Z*Z   tori zxz   : 57 isotypic components; rank modulo p of the span of the products = 6104 of 6561 (deficit 457)   [0.3s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 dim  81  group Z*Z   tori zxzx  : 57 isotypic components; rank modulo p of the span of the products = 6561 of 6561 (deficit 0)   [1.2s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 dim  81  group Z*Z   tori zxzxz : 57 isotypic components; rank modulo p of the span of the products = 6561 of 6561 (deficit 0)   [1.4s]
MOD p  V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 dim  81  group Z*Z   tori zxzxzx: 57 isotypic components; rank modulo p of the span of the products = 6561 of 6561 (deficit 0)   [2.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxz   : 60 isotypic components; rank modulo p of the span of the products = 5036 of 5184 (deficit 148)   [0.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzx  : 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzxz : 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.9s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzxzx: 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [1.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1 (x) z^-1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxz   : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1 (x) z^-1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzx  : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1 (x) z^-1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzxz : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.9s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^1 (x) V_1 (x) z^-1 (x) V_2 (x) z^-1 (x) V_1 dim  72  group Z*Z   tori zxzxzx: 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [1.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^2 (x) V_1 (x) z^-1 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxz   : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^2 (x) V_1 (x) z^-1 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzx  : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^2 (x) V_1 (x) z^-1 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzxz : 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.9s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^2 (x) V_1 (x) z^-1 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzxzx: 72 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [1.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxz   : 60 isotypic components; rank modulo p of the span of the products = 5036 of 5184 (deficit 148)   [0.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzx  : 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzxz : 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [0.9s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 (x) z^2 (x) V_2 (x) z^-2 (x) V_1 dim  72  group Z*Z   tori zxzxzx: 60 isotypic components; rank modulo p of the span of the products = 5184 of 5184 (deficit 0)   [1.3s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim 108  group Z*Z   tori zxz   : 69 isotypic components; rank modulo p of the span of the products = 10534 of 11664 (deficit 1130)   [1.0s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim 108  group Z*Z   tori zxzx  : 69 isotypic components; rank modulo p of the span of the products = 11664 of 11664 (deficit 0)   [5.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim 108  group Z*Z   tori zxzxz : 69 isotypic components; rank modulo p of the span of the products = 11664 of 11664 (deficit 0)   [4.7s]
MOD p  V_1 (x) z^1 (x) V_2 (x) z^-1 (x) V_2 (x) z^1 (x) V_2 (x) z^-1 (x) V_1 dim 108  group Z*Z   tori zxzxzx: 69 isotypic components; rank modulo p of the span of the products = 11664 of 11664 (deficit 0)   [9.5s]
