Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 541 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 87 + 64\cdot 541 + 514\cdot 541^{2} + 236\cdot 541^{3} + 462\cdot 541^{4} + 189\cdot 541^{5} + 197\cdot 541^{6} + 322\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 121 + 420\cdot 541 + 446\cdot 541^{2} + 87\cdot 541^{3} + 124\cdot 541^{4} + 264\cdot 541^{5} + 239\cdot 541^{6} + 157\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 246 + 55\cdot 541 + 454\cdot 541^{2} + 117\cdot 541^{3} + 133\cdot 541^{4} + 348\cdot 541^{5} + 280\cdot 541^{6} + 531\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 252 + 199\cdot 541 + 503\cdot 541^{2} + 56\cdot 541^{3} + 415\cdot 541^{4} + 504\cdot 541^{5} + 166\cdot 541^{6} + 463\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 290 + 341\cdot 541 + 37\cdot 541^{2} + 484\cdot 541^{3} + 125\cdot 541^{4} + 36\cdot 541^{5} + 374\cdot 541^{6} + 77\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 296 + 485\cdot 541 + 86\cdot 541^{2} + 423\cdot 541^{3} + 407\cdot 541^{4} + 192\cdot 541^{5} + 260\cdot 541^{6} + 9\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 421 + 120\cdot 541 + 94\cdot 541^{2} + 453\cdot 541^{3} + 416\cdot 541^{4} + 276\cdot 541^{5} + 301\cdot 541^{6} + 383\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 455 + 476\cdot 541 + 26\cdot 541^{2} + 304\cdot 541^{3} + 78\cdot 541^{4} + 351\cdot 541^{5} + 343\cdot 541^{6} + 218\cdot 541^{7} +O\left(541^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,7)(2,8)(3,4)(5,6)$ |
| $(1,8)(2,7)$ |
| $(1,3,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $8$ | $2$ | $(1,3)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)$ | $2$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,7)(4,5)$ | $-2$ |
| $8$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
| $8$ | $8$ | $(1,7,3,4,8,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
