Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 60\cdot 349 + 46\cdot 349^{2} + 52\cdot 349^{3} + 125\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 191\cdot 349 + 196\cdot 349^{2} + 72\cdot 349^{3} + 151\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 136\cdot 349 + 52\cdot 349^{2} + 88\cdot 349^{3} + 129\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 91\cdot 349 + 150\cdot 349^{2} + 124\cdot 349^{3} + 177\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 132 + 269\cdot 349 + 346\cdot 349^{2} + 146\cdot 349^{3} + 131\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 162 + 302\cdot 349 + 338\cdot 349^{2} + 327\cdot 349^{3} + 184\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 243 + 239\cdot 349 + 229\cdot 349^{2} + 17\cdot 349^{3} + 348\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 331 + 105\cdot 349 + 35\cdot 349^{2} + 217\cdot 349^{3} + 148\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,4)(2,5)(3,7)(6,8)$ |
| $(1,8,4,6)(2,7,5,3)$ |
| $(1,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,8,4,6)(2,7,5,3)$ | $0$ |
| $2$ | $8$ | $(1,2,6,3,4,5,8,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,8,2,4,7,6,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
