Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 449 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 131\cdot 449 + 253\cdot 449^{2} + 434\cdot 449^{3} + 243\cdot 449^{4} + 390\cdot 449^{5} + 306\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 162 + 344\cdot 449 + 210\cdot 449^{2} + 91\cdot 449^{3} + 211\cdot 449^{4} + 124\cdot 449^{5} + 417\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 202 + 442\cdot 449 + 318\cdot 449^{2} + 448\cdot 449^{3} + 236\cdot 449^{4} + 60\cdot 449^{5} + 37\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 291 + 54\cdot 449 + 54\cdot 449^{2} + 292\cdot 449^{3} + 79\cdot 449^{4} + 236\cdot 449^{5} + 237\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 352 + 341\cdot 449 + 45\cdot 449^{2} + 135\cdot 449^{3} + 217\cdot 449^{4} + 14\cdot 449^{5} + 136\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 381 + 65\cdot 449 + 287\cdot 449^{2} + 283\cdot 449^{3} + 198\cdot 449^{4} + 273\cdot 449^{5} + 229\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 395 + 40\cdot 449 + 439\cdot 449^{2} + 444\cdot 449^{3} + 344\cdot 449^{4} + 11\cdot 449^{5} + 263\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 429 + 374\cdot 449 + 186\cdot 449^{2} + 114\cdot 449^{3} + 263\cdot 449^{4} + 235\cdot 449^{5} + 168\cdot 449^{6} +O\left(449^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,2,3,6,4,5,8)$ |
| $(1,2,6,5)(3,7,8,4)$ |
| $(1,6)(4,7)$ |
| $(2,5)(3,8)$ |
| $(1,4,6,7)(2,8,5,3)$ |
| $(3,8)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-4$ |
| $2$ |
$2$ |
$(3,8)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(4,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,8)(3,5)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $8$ |
$2$ |
$(1,6)(3,4)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,5)(3,4,8,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,5)(3,7,8,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,6,7)(2,8,5,3)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,6,7)(2,3,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(3,7,8,4)$ |
$2$ |
| $4$ |
$4$ |
$(1,2,6,5)(3,8)(4,7)$ |
$-2$ |
| $8$ |
$8$ |
$(1,7,2,3,6,4,5,8)$ |
$0$ |
| $8$ |
$8$ |
$(1,7,2,8,6,4,5,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
