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 8.4.3902240000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.145.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.121945.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 3x^{6} - 15x^{4} - 29x^{2} + 29 \)
|
The roots of $f$ are computed in $\Q_{ 509 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 107 + 444\cdot 509 + 270\cdot 509^{2} + 308\cdot 509^{3} + 184\cdot 509^{4} + 119\cdot 509^{5} + 5\cdot 509^{6} + 205\cdot 509^{7} + 57\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 125 + 70\cdot 509 + 193\cdot 509^{2} + 376\cdot 509^{3} + 236\cdot 509^{4} + 119\cdot 509^{5} + 188\cdot 509^{6} + 211\cdot 509^{7} + 253\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 183 + 37\cdot 509 + 24\cdot 509^{2} + 458\cdot 509^{3} + 315\cdot 509^{4} + 466\cdot 509^{5} + 343\cdot 509^{6} + 288\cdot 509^{7} + 49\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 243 + 401\cdot 509 + 428\cdot 509^{2} + 158\cdot 509^{3} + 474\cdot 509^{4} + 413\cdot 509^{5} + 398\cdot 509^{6} + 449\cdot 509^{7} + 109\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 266 + 107\cdot 509 + 80\cdot 509^{2} + 350\cdot 509^{3} + 34\cdot 509^{4} + 95\cdot 509^{5} + 110\cdot 509^{6} + 59\cdot 509^{7} + 399\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 326 + 471\cdot 509 + 484\cdot 509^{2} + 50\cdot 509^{3} + 193\cdot 509^{4} + 42\cdot 509^{5} + 165\cdot 509^{6} + 220\cdot 509^{7} + 459\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 384 + 438\cdot 509 + 315\cdot 509^{2} + 132\cdot 509^{3} + 272\cdot 509^{4} + 389\cdot 509^{5} + 320\cdot 509^{6} + 297\cdot 509^{7} + 255\cdot 509^{8} +O(509^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 402 + 64\cdot 509 + 238\cdot 509^{2} + 200\cdot 509^{3} + 324\cdot 509^{4} + 389\cdot 509^{5} + 503\cdot 509^{6} + 303\cdot 509^{7} + 451\cdot 509^{8} +O(509^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ | ✓ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ | |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)$ | $\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,2,8,7)$ | $-\zeta_{4} + 1$ | |
| $2$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $\zeta_{4} - 1$ | |
| $2$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $-\zeta_{4} - 1$ | |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ | |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ | |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |