Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(512\)\(\medspace = 2^{9} \) |
| Artin stem field: | Galois closure of 8.0.134217728.3 |
| 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 4.2.2048.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{4} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 113 + 54\cdot 113^{2} + 102\cdot 113^{3} + 51\cdot 113^{4} + 95\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 90\cdot 113 + 11\cdot 113^{2} + 27\cdot 113^{3} + 56\cdot 113^{4} + 36\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 47\cdot 113 + 88\cdot 113^{2} + 57\cdot 113^{3} + 49\cdot 113^{4} + 43\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 74\cdot 113 + 9\cdot 113^{2} + 31\cdot 113^{3} + 9\cdot 113^{4} + 59\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 70 + 38\cdot 113 + 103\cdot 113^{2} + 81\cdot 113^{3} + 103\cdot 113^{4} + 53\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 + 65\cdot 113 + 24\cdot 113^{2} + 55\cdot 113^{3} + 63\cdot 113^{4} + 69\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 83 + 22\cdot 113 + 101\cdot 113^{2} + 85\cdot 113^{3} + 56\cdot 113^{4} + 76\cdot 113^{5} +O(113^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 111 + 108\cdot 113 + 58\cdot 113^{2} + 10\cdot 113^{3} + 61\cdot 113^{4} + 17\cdot 113^{5} +O(113^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(3,4,6,5)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(3,5,6,4)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,7)(3,4,6,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $0$ |
| $4$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.