Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 349 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 218\cdot 349 + 198\cdot 349^{2} + 31\cdot 349^{3} + 93\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 93 + 113\cdot 349 + 87\cdot 349^{2} + 36\cdot 349^{3} + 177\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 132 + 251\cdot 349 + 97\cdot 349^{2} + 177\cdot 349^{3} + 121\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 171 + 260\cdot 349 + 315\cdot 349^{2} + 283\cdot 349^{3} + 3\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 185 + 148\cdot 349 + 183\cdot 349^{2} + 130\cdot 349^{3} + 193\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 192 + 290\cdot 349 + 227\cdot 349^{2} + 249\cdot 349^{3} + 318\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 267 + 211\cdot 349 + 192\cdot 349^{2} + 193\cdot 349^{3} + 29\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 334 + 250\cdot 349 + 92\cdot 349^{2} + 293\cdot 349^{3} + 109\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,7)(3,5)(6,8)$ |
| $(1,2,4,7)(3,8,6,5)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,6,7,8,4,3,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(2,7)(3,5)(6,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,2)(3,5,6,8)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,8,4,5)(2,3,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,7,8,4,3,2,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,7,5,4,6,2,8)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
