Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(1013888\)\(\medspace = 2^{7} \cdot 89^{2} \) |
| Artin stem field: | Galois closure of 8.0.2956854296576.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | even |
| Determinant: | 1.712.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.4.22559008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 16x^{6} + 28x^{5} + 164x^{4} - 60x^{3} - 56x^{2} + 76x + 313 \)
|
The roots of $f$ are computed in $\Q_{ 367 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 45 + 166\cdot 367 + 102\cdot 367^{2} + 97\cdot 367^{3} + 20\cdot 367^{4} + 355\cdot 367^{5} + 183\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 + 203\cdot 367 + 335\cdot 367^{2} + 317\cdot 367^{3} + 340\cdot 367^{4} + 367^{5} + 299\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 91 + 23\cdot 367 + 294\cdot 367^{2} + 95\cdot 367^{3} + 337\cdot 367^{4} + 247\cdot 367^{5} + 130\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 98 + 177\cdot 367 + 304\cdot 367^{2} + 192\cdot 367^{3} + 175\cdot 367^{4} + 257\cdot 367^{5} + 71\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 127 + 243\cdot 367 + 130\cdot 367^{2} + 322\cdot 367^{3} + 180\cdot 367^{4} + 162\cdot 367^{5} + 63\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 173 + 178\cdot 367 + 276\cdot 367^{2} + 264\cdot 367^{3} + 174\cdot 367^{4} + 265\cdot 367^{5} + 149\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 211 + 91\cdot 367 + 40\cdot 367^{2} + 335\cdot 367^{3} + 160\cdot 367^{4} + 104\cdot 367^{5} + 265\cdot 367^{6} +O(367^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 296 + 17\cdot 367 + 351\cdot 367^{2} + 208\cdot 367^{3} + 77\cdot 367^{4} + 73\cdot 367^{5} + 304\cdot 367^{6} +O(367^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(2,5)(3,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,6,4)(2,8,5,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,8,5,3)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(2,3,5,8)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6)(2,3,5,8)(4,7)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6)(2,8,5,3)(4,7)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,4,6,7)(2,8,5,3)$ | $0$ |
| $4$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $0$ |
| $4$ | $8$ | $(1,2,4,3,6,5,7,8)$ | $0$ |
| $4$ | $8$ | $(1,3,7,2,6,8,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.