Basic invariants
| Dimension: | $4$ |
| Group: | $Z_8 : Z_8^\times$ |
| Conductor: | \(21233664\)\(\medspace = 2^{18} \cdot 3^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.97844723712.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Z_8 : Z_8^\times$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2\times D_4$ |
| Projective stem field: | Galois closure of 8.0.21233664.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} + 66x^{4} + 72x^{2} + 18 \)
|
The roots of $f$ are computed in $\Q_{ 313 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 135\cdot 313 + 300\cdot 313^{2} + 184\cdot 313^{3} + 118\cdot 313^{4} + 14\cdot 313^{5} + 166\cdot 313^{6} + 127\cdot 313^{7} + 27\cdot 313^{8} + 131\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 67 + 71\cdot 313 + 9\cdot 313^{2} + 107\cdot 313^{3} + 167\cdot 313^{4} + 57\cdot 313^{5} + 96\cdot 313^{6} + 182\cdot 313^{7} + 5\cdot 313^{8} + 45\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 99 + 11\cdot 313 + 185\cdot 313^{2} + 272\cdot 313^{3} + 156\cdot 313^{4} + 163\cdot 313^{5} + 234\cdot 313^{6} + 21\cdot 313^{7} + 61\cdot 313^{8} + 287\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 135 + 311\cdot 313 + 73\cdot 313^{2} + 305\cdot 313^{3} + 204\cdot 313^{4} + 108\cdot 313^{5} + 222\cdot 313^{6} + 36\cdot 313^{7} + 275\cdot 313^{8} + 16\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 178 + 313 + 239\cdot 313^{2} + 7\cdot 313^{3} + 108\cdot 313^{4} + 204\cdot 313^{5} + 90\cdot 313^{6} + 276\cdot 313^{7} + 37\cdot 313^{8} + 296\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 214 + 301\cdot 313 + 127\cdot 313^{2} + 40\cdot 313^{3} + 156\cdot 313^{4} + 149\cdot 313^{5} + 78\cdot 313^{6} + 291\cdot 313^{7} + 251\cdot 313^{8} + 25\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 246 + 241\cdot 313 + 303\cdot 313^{2} + 205\cdot 313^{3} + 145\cdot 313^{4} + 255\cdot 313^{5} + 216\cdot 313^{6} + 130\cdot 313^{7} + 307\cdot 313^{8} + 267\cdot 313^{9} +O(313^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 306 + 177\cdot 313 + 12\cdot 313^{2} + 128\cdot 313^{3} + 194\cdot 313^{4} + 298\cdot 313^{5} + 146\cdot 313^{6} + 185\cdot 313^{7} + 285\cdot 313^{8} + 181\cdot 313^{9} +O(313^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ | |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ | |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ | |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ | ✓ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $4$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | |
| $4$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $0$ | |
| $4$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $0$ |