Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1024\)\(\medspace = 2^{10} \) |
| Artin stem field: | Galois closure of 8.0.2147483648.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.16.4t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.512.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} + 24x^{4} - 32x^{2} + 18 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 78\cdot 113 + 76\cdot 113^{2} + 9\cdot 113^{3} + 29\cdot 113^{4} + 22\cdot 113^{5} + 33\cdot 113^{6} + 48\cdot 113^{7} + 38\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 17\cdot 113 + 9\cdot 113^{2} + 68\cdot 113^{3} + 26\cdot 113^{4} + 107\cdot 113^{5} + 90\cdot 113^{6} + 102\cdot 113^{7} +O(113^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 13\cdot 113 + 82\cdot 113^{2} + 6\cdot 113^{3} + 42\cdot 113^{4} + 91\cdot 113^{5} + 107\cdot 113^{6} + 50\cdot 113^{7} + 104\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 + 95\cdot 113 + 7\cdot 113^{2} + 54\cdot 113^{3} + 87\cdot 113^{4} + 112\cdot 113^{5} + 33\cdot 113^{6} + 80\cdot 113^{7} + 80\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 82 + 17\cdot 113 + 105\cdot 113^{2} + 58\cdot 113^{3} + 25\cdot 113^{4} + 79\cdot 113^{6} + 32\cdot 113^{7} + 32\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 + 99\cdot 113 + 30\cdot 113^{2} + 106\cdot 113^{3} + 70\cdot 113^{4} + 21\cdot 113^{5} + 5\cdot 113^{6} + 62\cdot 113^{7} + 8\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 104 + 95\cdot 113 + 103\cdot 113^{2} + 44\cdot 113^{3} + 86\cdot 113^{4} + 5\cdot 113^{5} + 22\cdot 113^{6} + 10\cdot 113^{7} + 112\cdot 113^{8} +O(113^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 107 + 34\cdot 113 + 36\cdot 113^{2} + 103\cdot 113^{3} + 83\cdot 113^{4} + 90\cdot 113^{5} + 79\cdot 113^{6} + 64\cdot 113^{7} + 74\cdot 113^{8} +O(113^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2,8,7)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $0$ |
| $4$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.