Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.45556992.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.156.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.105456.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 7x^{6} + 20x^{4} + 26x^{2} + 13 \)
|
The roots of $f$ are computed in $\Q_{ 751 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 53 + 196\cdot 751 + 731\cdot 751^{2} + 718\cdot 751^{3} + 530\cdot 751^{4} + 33\cdot 751^{5} + 495\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 129 + 18\cdot 751 + 363\cdot 751^{2} + 103\cdot 751^{3} + 63\cdot 751^{4} + 477\cdot 751^{5} + 741\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 203 + 411\cdot 751 + 393\cdot 751^{2} + 421\cdot 751^{3} + 62\cdot 751^{4} + 338\cdot 751^{5} + 130\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 228 + 217\cdot 751 + 685\cdot 751^{2} + 15\cdot 751^{3} + 427\cdot 751^{4} + 585\cdot 751^{5} + 699\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 523 + 533\cdot 751 + 65\cdot 751^{2} + 735\cdot 751^{3} + 323\cdot 751^{4} + 165\cdot 751^{5} + 51\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 548 + 339\cdot 751 + 357\cdot 751^{2} + 329\cdot 751^{3} + 688\cdot 751^{4} + 412\cdot 751^{5} + 620\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 622 + 732\cdot 751 + 387\cdot 751^{2} + 647\cdot 751^{3} + 687\cdot 751^{4} + 273\cdot 751^{5} + 9\cdot 751^{6} +O(751^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 698 + 554\cdot 751 + 19\cdot 751^{2} + 32\cdot 751^{3} + 220\cdot 751^{4} + 717\cdot 751^{5} + 255\cdot 751^{6} +O(751^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(3,5,6,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(3,4,6,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,2,8,7)(3,6)(4,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
| $4$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.