Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Conductor: | \(80714907\)\(\medspace = 3^{3} \cdot 7^{2} \cdot 13^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.4546939761.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_2^2\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.242144721.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 41x^{4} - 38x^{3} - 64x^{2} + 58x + 53 \)
|
The roots of $f$ are computed in $\Q_{ 523 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 40\cdot 523 + 130\cdot 523^{2} + 522\cdot 523^{3} + 4\cdot 523^{4} + 470\cdot 523^{5} + 262\cdot 523^{6} + 389\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 + 374\cdot 523 + 20\cdot 523^{2} + 370\cdot 523^{3} + 420\cdot 523^{4} + 432\cdot 523^{5} + 448\cdot 523^{6} + 77\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 117 + 195\cdot 523 + 54\cdot 523^{2} + 358\cdot 523^{3} + 474\cdot 523^{4} + 55\cdot 523^{5} + 433\cdot 523^{6} + 286\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 124 + 149\cdot 523 + 153\cdot 523^{2} + 408\cdot 523^{3} + 280\cdot 523^{4} + 43\cdot 523^{5} + 44\cdot 523^{6} + 71\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 153 + 428\cdot 523 + 484\cdot 523^{2} + 415\cdot 523^{3} + 338\cdot 523^{4} + 367\cdot 523^{5} + 153\cdot 523^{6} + 304\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 375 + 469\cdot 523 + 447\cdot 523^{2} + 343\cdot 523^{3} + 348\cdot 523^{4} + 417\cdot 523^{5} + 427\cdot 523^{6} + 195\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 380 + 64\cdot 523 + 510\cdot 523^{2} + 253\cdot 523^{3} + 271\cdot 523^{4} + 12\cdot 523^{5} + 238\cdot 523^{6} + 141\cdot 523^{7} +O(523^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 381 + 370\cdot 523 + 290\cdot 523^{2} + 465\cdot 523^{3} + 474\cdot 523^{4} + 291\cdot 523^{5} + 83\cdot 523^{6} + 102\cdot 523^{7} +O(523^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $-4$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(3,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $2$ |
| $4$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(6,8)$ | $-2$ |
| $4$ | $4$ | $(1,5,8,7)(2,3,4,6)$ | $0$ |
| $4$ | $4$ | $(1,5,8,7)(2,6,4,3)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,7,4,5)$ | $0$ |
| $8$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $0$ |
| $8$ | $4$ | $(1,6,8,3)(5,7)$ | $0$ |
| $8$ | $4$ | $(1,2,6,7)(3,5,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.