Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 449 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 128\cdot 449 + 36\cdot 449^{2} + 388\cdot 449^{3} + 373\cdot 449^{4} + 58\cdot 449^{5} + 437\cdot 449^{6} + 150\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 98 + 83\cdot 449 + 170\cdot 449^{2} + 40\cdot 449^{3} + 309\cdot 449^{4} + 296\cdot 449^{5} + 233\cdot 449^{6} + 72\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 100 + 131\cdot 449 + 366\cdot 449^{2} + 344\cdot 449^{3} + 354\cdot 449^{4} + 222\cdot 449^{5} + 235\cdot 449^{6} + 17\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 188 + 198\cdot 449 + 417\cdot 449^{2} + 386\cdot 449^{3} + 209\cdot 449^{4} + 445\cdot 449^{5} + 89\cdot 449^{6} + 206\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 291 + 235\cdot 449 + 379\cdot 449^{2} + 108\cdot 449^{3} + 212\cdot 449^{4} + 357\cdot 449^{5} + 101\cdot 449^{6} + 128\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 319 + 332\cdot 449 + 183\cdot 449^{2} + 57\cdot 449^{3} + 121\cdot 449^{4} + 321\cdot 449^{5} + 21\cdot 449^{6} + 97\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 362 + 335\cdot 449 + 64\cdot 449^{2} + 14\cdot 449^{3} + 102\cdot 449^{4} + 36\cdot 449^{5} + 269\cdot 449^{6} + 412\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 381 + 350\cdot 449 + 177\cdot 449^{2} + 6\cdot 449^{3} + 113\cdot 449^{4} + 57\cdot 449^{5} + 407\cdot 449^{6} + 261\cdot 449^{7} +O\left(449^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,4)(2,6,8,3)$ |
| $(1,7)(3,6)$ |
| $(3,6)(4,5)$ |
| $(1,5,3,8,7,4,6,2)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(2,8)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $8$ | $2$ | $(2,4)(3,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,6)(2,5,8,4)$ | $0$ |
| $2$ | $4$ | $(1,6,7,3)(2,5,8,4)$ | $0$ |
| $4$ | $4$ | $(1,5,7,4)(2,6,8,3)$ | $0$ |
| $4$ | $4$ | $(2,4,8,5)$ | $-2$ |
| $4$ | $4$ | $(1,4,7,5)(2,6,8,3)$ | $0$ |
| $4$ | $4$ | $(1,7)(2,4,8,5)(3,6)$ | $2$ |
| $8$ | $8$ | $(1,5,3,8,7,4,6,2)$ | $0$ |
| $8$ | $8$ | $(1,5,6,8,7,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
