Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(928\)\(\medspace = 2^{5} \cdot 29 \) |
| Artin stem field: | Galois closure of 8.0.1598357504.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.29.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.390224.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 10x^{4} + 29 \)
|
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 150\cdot 181 + 33\cdot 181^{2} + 98\cdot 181^{3} + 4\cdot 181^{5} + 120\cdot 181^{6} + 54\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 110\cdot 181 + 40\cdot 181^{2} + 142\cdot 181^{3} + 62\cdot 181^{4} + 130\cdot 181^{5} + 117\cdot 181^{6} + 58\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 65 + 22\cdot 181 + 64\cdot 181^{2} + 163\cdot 181^{3} + 53\cdot 181^{4} + 88\cdot 181^{5} + 167\cdot 181^{6} + 82\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 79 + 155\cdot 181 + 56\cdot 181^{2} + 96\cdot 181^{3} + 167\cdot 181^{4} + 37\cdot 181^{5} + 120\cdot 181^{6} + 26\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 102 + 25\cdot 181 + 124\cdot 181^{2} + 84\cdot 181^{3} + 13\cdot 181^{4} + 143\cdot 181^{5} + 60\cdot 181^{6} + 154\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 116 + 158\cdot 181 + 116\cdot 181^{2} + 17\cdot 181^{3} + 127\cdot 181^{4} + 92\cdot 181^{5} + 13\cdot 181^{6} + 98\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 128 + 70\cdot 181 + 140\cdot 181^{2} + 38\cdot 181^{3} + 118\cdot 181^{4} + 50\cdot 181^{5} + 63\cdot 181^{6} + 122\cdot 181^{7} +O(181^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 149 + 30\cdot 181 + 147\cdot 181^{2} + 82\cdot 181^{3} + 180\cdot 181^{4} + 176\cdot 181^{5} + 60\cdot 181^{6} + 126\cdot 181^{7} +O(181^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(2,4,7,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,5,7,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $0$ |
| $4$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.