Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Conductor: | \(508805\)\(\medspace = 5 \cdot 11^{2} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.307827025.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $C_2^2\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.63600625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 3x^{6} - 8x^{5} + 9x^{4} - 8x^{3} + 16x^{2} - 20x + 9 \)
|
The roots of $f$ are computed in $\Q_{ 929 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 345\cdot 929 + 280\cdot 929^{2} + 51\cdot 929^{3} + 355\cdot 929^{4} + 762\cdot 929^{5} + 249\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 140 + 502\cdot 929 + 7\cdot 929^{2} + 62\cdot 929^{3} + 107\cdot 929^{4} + 18\cdot 929^{5} + 279\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 151 + 749\cdot 929 + 698\cdot 929^{2} + 11\cdot 929^{3} + 466\cdot 929^{4} + 220\cdot 929^{5} + 41\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 591 + 857\cdot 929 + 639\cdot 929^{2} + 212\cdot 929^{3} + 823\cdot 929^{4} + 223\cdot 929^{5} + 250\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 595 + 595\cdot 929 + 818\cdot 929^{2} + 210\cdot 929^{3} + 452\cdot 929^{4} + 756\cdot 929^{5} + 702\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 676 + 630\cdot 929 + 522\cdot 929^{2} + 62\cdot 929^{3} + 287\cdot 929^{4} + 283\cdot 929^{5} + 254\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 740 + 158\cdot 929 + 686\cdot 929^{2} + 102\cdot 929^{3} + 767\cdot 929^{4} + 370\cdot 929^{5} + 441\cdot 929^{6} +O(929^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 793 + 805\cdot 929 + 61\cdot 929^{2} + 215\cdot 929^{3} + 458\cdot 929^{4} + 151\cdot 929^{5} + 568\cdot 929^{6} +O(929^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $-4$ |
| $2$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(2,4)(7,8)$ | $2$ |
| $4$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,7,3,2)(4,5,8,6)$ | $0$ |
| $4$ | $4$ | $(1,5,3,6)(2,8,7,4)$ | $0$ |
| $4$ | $4$ | $(1,2,3,7)(4,5,8,6)$ | $0$ |
| $8$ | $4$ | $(1,7,6,8)(2,5,4,3)$ | $0$ |
| $8$ | $4$ | $(1,7,5,8)(2,6,4,3)$ | $0$ |
| $8$ | $4$ | $(2,8,7,4)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.