Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Artin stem field: | Galois closure of 8.4.5151653888.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.136.4t1.b.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.39304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 12x^{5} - 12x^{4} + 8x^{3} + 16x^{2} + 12x - 7 \)
|
The roots of $f$ are computed in $\Q_{ 383 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 34\cdot 383 + 355\cdot 383^{2} + 307\cdot 383^{3} + 169\cdot 383^{4} + 265\cdot 383^{5} + 32\cdot 383^{6} + 263\cdot 383^{7} + 369\cdot 383^{8} + 238\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 111 + 107\cdot 383 + 257\cdot 383^{2} + 174\cdot 383^{3} + 74\cdot 383^{4} + 76\cdot 383^{5} + 217\cdot 383^{6} + 191\cdot 383^{7} + 209\cdot 383^{8} + 9\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 160 + 17\cdot 383 + 249\cdot 383^{2} + 138\cdot 383^{3} + 25\cdot 383^{4} + 259\cdot 383^{5} + 132\cdot 383^{6} + 368\cdot 383^{7} + 22\cdot 383^{8} + 311\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 265 + 349\cdot 383 + 280\cdot 383^{2} + 22\cdot 383^{3} + 29\cdot 383^{4} + 121\cdot 383^{5} + 243\cdot 383^{6} + 161\cdot 383^{7} + 278\cdot 383^{8} + 289\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 311 + 253\cdot 383 + 221\cdot 383^{2} + 153\cdot 383^{3} + 231\cdot 383^{4} + 84\cdot 383^{5} + 136\cdot 383^{6} + 100\cdot 383^{7} + 324\cdot 383^{8} + 116\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 318 + 370\cdot 383 + 314\cdot 383^{2} + 129\cdot 383^{3} + 290\cdot 383^{4} + 339\cdot 383^{5} + 379\cdot 383^{6} + 210\cdot 383^{7} + 245\cdot 383^{8} + 17\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 360 + 123\cdot 383 + 363\cdot 383^{2} + 343\cdot 383^{3} + 218\cdot 383^{4} + 82\cdot 383^{5} + 117\cdot 383^{6} + 86\cdot 383^{7} + 173\cdot 383^{8} + 320\cdot 383^{9} +O(383^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 364 + 274\cdot 383 + 255\cdot 383^{2} + 260\cdot 383^{3} + 109\cdot 383^{4} + 303\cdot 383^{5} + 272\cdot 383^{6} + 149\cdot 383^{7} + 291\cdot 383^{8} + 227\cdot 383^{9} +O(383^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,7,2,3)(4,6,8,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,7)(4,5,8,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(4,6,8,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(4,5,8,6)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,7,2,3)(4,8)(5,6)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,2,7)(4,8)(5,6)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,7,2,3)(4,5,8,6)$ | $0$ |
| $4$ | $4$ | $(1,5,2,6)(3,8,7,4)$ | $0$ |
| $4$ | $8$ | $(1,8,7,5,2,4,3,6)$ | $0$ |
| $4$ | $8$ | $(1,5,3,8,2,6,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.