Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 + 2\cdot 349 + 133\cdot 349^{2} + 245\cdot 349^{3} + 103\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 103 + 157\cdot 349 + 95\cdot 349^{2} + 50\cdot 349^{3} + 223\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 132 + 320\cdot 349 + 8\cdot 349^{2} + 148\cdot 349^{3} + 84\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 168 + 195\cdot 349 + 177\cdot 349^{2} + 217\cdot 349^{3} + 23\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 197 + 333\cdot 349 + 263\cdot 349^{2} + 172\cdot 349^{3} + 27\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 209 + 185\cdot 349 + 140\cdot 349^{2} + 36\cdot 349^{3} + 220\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 231 + 11\cdot 349 + 161\cdot 349^{2} + 257\cdot 349^{3} + 74\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 320 + 189\cdot 349 + 66\cdot 349^{2} + 268\cdot 349^{3} + 289\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(2,7)(4,5)$ |
| $(1,8)(2,5)(3,6)(4,7)$ |
| $(2,5,7,4)(6,8)$ |
| $(1,2,3,7)(4,8,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $4$ | $4$ | $(1,2,3,7)(4,8,5,6)$ | $0$ |
| $4$ | $4$ | $(1,5,8,7)(2,3,4,6)$ | $0$ |
| $4$ | $4$ | $(1,7,8,5)(2,6,4,3)$ | $0$ |
| $4$ | $4$ | $(2,5,7,4)(6,8)$ | $0$ |
| $4$ | $4$ | $(2,4,7,5)(6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
