Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 113\cdot 409 + 211\cdot 409^{2} + 87\cdot 409^{3} + 82\cdot 409^{4} + 65\cdot 409^{5} + 272\cdot 409^{6} + 224\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 361\cdot 409 + 111\cdot 409^{2} + 385\cdot 409^{3} + 121\cdot 409^{4} + 327\cdot 409^{5} + 245\cdot 409^{6} + 146\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 + 210\cdot 409 + 77\cdot 409^{2} + 72\cdot 409^{3} + 141\cdot 409^{4} + 4\cdot 409^{5} + 79\cdot 409^{6} + 123\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 74 + 405\cdot 409 + 166\cdot 409^{2} + 15\cdot 409^{3} + 137\cdot 409^{4} + 301\cdot 409^{5} + 121\cdot 409^{6} + 352\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 335 + 3\cdot 409 + 242\cdot 409^{2} + 393\cdot 409^{3} + 271\cdot 409^{4} + 107\cdot 409^{5} + 287\cdot 409^{6} + 56\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 357 + 198\cdot 409 + 331\cdot 409^{2} + 336\cdot 409^{3} + 267\cdot 409^{4} + 404\cdot 409^{5} + 329\cdot 409^{6} + 285\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 386 + 47\cdot 409 + 297\cdot 409^{2} + 23\cdot 409^{3} + 287\cdot 409^{4} + 81\cdot 409^{5} + 163\cdot 409^{6} + 262\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 392 + 295\cdot 409 + 197\cdot 409^{2} + 321\cdot 409^{3} + 326\cdot 409^{4} + 343\cdot 409^{5} + 136\cdot 409^{6} + 184\cdot 409^{7} +O\left(409^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(3,6)(4,5)$ |
| $(2,7)(3,5)(4,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $0$ |
| $4$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
