Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 + 259\cdot 457 + 207\cdot 457^{2} + 26\cdot 457^{3} + 258\cdot 457^{4} + 4\cdot 457^{5} + 348\cdot 457^{6} + 398\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 + 182\cdot 457 + 213\cdot 457^{2} + 328\cdot 457^{3} + 249\cdot 457^{4} + 275\cdot 457^{5} + 411\cdot 457^{6} + 94\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 143 + 287\cdot 457 + 316\cdot 457^{2} + 78\cdot 457^{3} + 8\cdot 457^{4} + 82\cdot 457^{5} + 202\cdot 457^{6} + 173\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 153 + 211\cdot 457 + 12\cdot 457^{2} + 320\cdot 457^{3} + 303\cdot 457^{4} + 367\cdot 457^{5} + 371\cdot 457^{6} + 306\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 167 + 117\cdot 457 + 262\cdot 457^{2} + 32\cdot 457^{3} + 376\cdot 457^{4} + 390\cdot 457^{5} + 26\cdot 457^{6} + 403\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 230 + 438\cdot 457 + 123\cdot 457^{2} + 305\cdot 457^{3} + 251\cdot 457^{4} + 249\cdot 457^{5} + 353\cdot 457^{6} + 316\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 258 + 75\cdot 457 + 329\cdot 457^{2} + 180\cdot 457^{3} + 35\cdot 457^{4} + 81\cdot 457^{5} + 71\cdot 457^{6} + 316\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 310 + 256\cdot 457 + 362\cdot 457^{2} + 98\cdot 457^{3} + 345\cdot 457^{4} + 376\cdot 457^{5} + 42\cdot 457^{6} + 275\cdot 457^{7} +O\left(457^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,5,4)(2,8,7,3)$ |
| $(3,8)(4,6)$ |
| $(2,7)(3,8)$ |
| $(1,5)(4,6)$ |
| $(1,4,2,8,5,6,7,3)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $0$ |
| $8$ | $2$ | $(1,7)(2,5)(3,8)$ | $0$ |
| $2$ | $4$ | $(1,2,5,7)(3,4,8,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,7)(3,6,8,4)$ | $0$ |
| $4$ | $4$ | $(1,6,5,4)(2,8,7,3)$ | $0$ |
| $4$ | $4$ | $(1,6,5,4)(2,3,7,8)$ | $0$ |
| $4$ | $4$ | $(1,7,5,2)$ | $2$ |
| $4$ | $4$ | $(1,7,5,2)(3,8)(4,6)$ | $-2$ |
| $8$ | $8$ | $(1,4,2,8,5,6,7,3)$ | $0$ |
| $8$ | $8$ | $(1,4,2,3,5,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
