Basic invariants
| Dimension: | $4$ |
| Group: | $C_2^3 : C_4 $ |
| Conductor: | \(1275125\)\(\medspace = 5^{3} \cdot 101^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1625943765625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^3 : C_4 $ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $C_2^2:C_4$ |
| Projective stem field: | Galois closure of 8.0.159390625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 10x^{5} + 31x^{4} - 50x^{3} + 47x^{2} - 23x + 6 \)
|
The roots of $f$ are computed in $\Q_{ 701 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 29\cdot 701 + 368\cdot 701^{2} + 176\cdot 701^{3} + 376\cdot 701^{4} + 215\cdot 701^{5} + 164\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 254 + 118\cdot 701 + 550\cdot 701^{2} + 255\cdot 701^{3} + 493\cdot 701^{4} + 666\cdot 701^{5} + 652\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 259 + 321\cdot 701 + 359\cdot 701^{2} + 326\cdot 701^{3} + 440\cdot 701^{4} + 579\cdot 701^{5} + 505\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 302 + 680\cdot 701 + 190\cdot 701^{2} + 226\cdot 701^{3} + 211\cdot 701^{4} + 321\cdot 701^{5} + 621\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 400 + 20\cdot 701 + 510\cdot 701^{2} + 474\cdot 701^{3} + 489\cdot 701^{4} + 379\cdot 701^{5} + 79\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 443 + 379\cdot 701 + 341\cdot 701^{2} + 374\cdot 701^{3} + 260\cdot 701^{4} + 121\cdot 701^{5} + 195\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 448 + 582\cdot 701 + 150\cdot 701^{2} + 445\cdot 701^{3} + 207\cdot 701^{4} + 34\cdot 701^{5} + 48\cdot 701^{6} +O(701^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 671 + 671\cdot 701 + 332\cdot 701^{2} + 524\cdot 701^{3} + 324\cdot 701^{4} + 485\cdot 701^{5} + 536\cdot 701^{6} +O(701^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $4$ | $4$ | $(1,5,7,3)(2,6,8,4)$ | $0$ |
| $4$ | $4$ | $(1,3,7,5)(2,4,8,6)$ | $0$ |
| $4$ | $4$ | $(1,8)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,8)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.