Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 32\cdot 337 + 251\cdot 337^{2} + 21\cdot 337^{3} + 316\cdot 337^{4} + 325\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 271\cdot 337 + 65\cdot 337^{2} + 297\cdot 337^{3} + 279\cdot 337^{4} + 116\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 113 + 290\cdot 337 + 244\cdot 337^{2} + 57\cdot 337^{3} + 312\cdot 337^{4} + 320\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 124 + 156\cdot 337 + 320\cdot 337^{2} + 17\cdot 337^{3} + 193\cdot 337^{4} + 303\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 213 + 180\cdot 337 + 16\cdot 337^{2} + 319\cdot 337^{3} + 143\cdot 337^{4} + 33\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 224 + 46\cdot 337 + 92\cdot 337^{2} + 279\cdot 337^{3} + 24\cdot 337^{4} + 16\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 296 + 65\cdot 337 + 271\cdot 337^{2} + 39\cdot 337^{3} + 57\cdot 337^{4} + 220\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 308 + 304\cdot 337 + 85\cdot 337^{2} + 315\cdot 337^{3} + 20\cdot 337^{4} + 11\cdot 337^{5} +O\left(337^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,8)(4,5)$ |
| $(1,2,4,3)(5,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,4,3)(5,6,8,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,4,2)(5,7,8,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,8,3)(2,5,7,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,4)(3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
