Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(2320\)\(\medspace = 2^{4} \cdot 5 \cdot 29 \) |
| Artin stem field: | Galois closure of 8.4.22632992000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.145.4t1.b.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.3625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} - 18x^{4} - 15x^{2} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 431 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 315\cdot 431^{2} + 16\cdot 431^{3} + 196\cdot 431^{4} + 419\cdot 431^{5} + 417\cdot 431^{6} + 297\cdot 431^{7} + 95\cdot 431^{8} + 177\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 176 + 250\cdot 431 + 170\cdot 431^{2} + 176\cdot 431^{3} + 337\cdot 431^{4} + 96\cdot 431^{5} + 408\cdot 431^{6} + 277\cdot 431^{7} + 347\cdot 431^{8} + 393\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 179 + 101\cdot 431 + 288\cdot 431^{2} + 161\cdot 431^{3} + 233\cdot 431^{4} + 382\cdot 431^{5} + 291\cdot 431^{6} + 226\cdot 431^{7} + 360\cdot 431^{8} + 67\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 184 + 93\cdot 431 + 214\cdot 431^{2} + 380\cdot 431^{3} + 203\cdot 431^{4} + 396\cdot 431^{5} + 360\cdot 431^{6} + 36\cdot 431^{7} + 185\cdot 431^{8} + 419\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 247 + 337\cdot 431 + 216\cdot 431^{2} + 50\cdot 431^{3} + 227\cdot 431^{4} + 34\cdot 431^{5} + 70\cdot 431^{6} + 394\cdot 431^{7} + 245\cdot 431^{8} + 11\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 252 + 329\cdot 431 + 142\cdot 431^{2} + 269\cdot 431^{3} + 197\cdot 431^{4} + 48\cdot 431^{5} + 139\cdot 431^{6} + 204\cdot 431^{7} + 70\cdot 431^{8} + 363\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 255 + 180\cdot 431 + 260\cdot 431^{2} + 254\cdot 431^{3} + 93\cdot 431^{4} + 334\cdot 431^{5} + 22\cdot 431^{6} + 153\cdot 431^{7} + 83\cdot 431^{8} + 37\cdot 431^{9} +O(431^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 408 + 430\cdot 431 + 115\cdot 431^{2} + 414\cdot 431^{3} + 234\cdot 431^{4} + 11\cdot 431^{5} + 13\cdot 431^{6} + 133\cdot 431^{7} + 335\cdot 431^{8} + 253\cdot 431^{9} +O(431^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ | ✓ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ | |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(2,3,7,6)$ | $\zeta_{4} + 1$ | |
| $2$ | $4$ | $(2,6,7,3)$ | $-\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $-\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,8)(2,3,7,6)(4,5)$ | $\zeta_{4} - 1$ | |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $4$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ | |
| $4$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |