Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Artin number field: | Galois closure of 8.0.20123648.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.39304.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 38 + 166\cdot 307 + 69\cdot 307^{2} + 158\cdot 307^{3} + 143\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 50 + 75\cdot 307 + 241\cdot 307^{2} + 13\cdot 307^{3} + 63\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 75 + 64\cdot 307 + 64\cdot 307^{2} + 257\cdot 307^{3} + 69\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 84 + 92\cdot 307 + 186\cdot 307^{2} + 31\cdot 307^{3} + 223\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 95 + 238\cdot 307 + 137\cdot 307^{2} + 64\cdot 307^{3} + 92\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 148 + 267\cdot 307 + 168\cdot 307^{2} + 146\cdot 307^{3} + 65\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 196 + 34\cdot 307 + 212\cdot 307^{2} + 262\cdot 307^{3} + 29\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 237 + 289\cdot 307 + 147\cdot 307^{2} + 293\cdot 307^{3} + 233\cdot 307^{4} +O(307^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,4)(6,8)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,8,4,6)(2,7,5,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,4,8)(2,3,5,7)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,4,6)$ | $-\zeta_{4} + 1$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,4,8)$ | $\zeta_{4} + 1$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,4)(2,7,5,3)(6,8)$ | $-\zeta_{4} - 1$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,4)(2,3,5,7)(6,8)$ | $\zeta_{4} - 1$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,4,8)(2,7,5,3)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,2,4,5)(3,8,7,6)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,2,8,7,4,5,6,3)$ | $0$ | $0$ |
| $4$ | $8$ | $(1,7,6,2,4,3,8,5)$ | $0$ | $0$ |