Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 353 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 34 + 85\cdot 353 + 283\cdot 353^{2} + 180\cdot 353^{3} + 202\cdot 353^{4} + 130\cdot 353^{5} + 207\cdot 353^{6} + 93\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 243\cdot 353 + 43\cdot 353^{2} + 269\cdot 353^{3} + 273\cdot 353^{4} + 72\cdot 353^{5} + 111\cdot 353^{6} + 104\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 78 + 249\cdot 353 + 318\cdot 353^{2} + 125\cdot 353^{3} + 334\cdot 353^{4} + 283\cdot 353^{5} + 349\cdot 353^{6} + 21\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 123 + 291\cdot 353 + 73\cdot 353^{2} + 251\cdot 353^{3} + 254\cdot 353^{4} + 176\cdot 353^{5} + 77\cdot 353^{6} + 145\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 230 + 61\cdot 353 + 279\cdot 353^{2} + 101\cdot 353^{3} + 98\cdot 353^{4} + 176\cdot 353^{5} + 275\cdot 353^{6} + 207\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 275 + 103\cdot 353 + 34\cdot 353^{2} + 227\cdot 353^{3} + 18\cdot 353^{4} + 69\cdot 353^{5} + 3\cdot 353^{6} + 331\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 301 + 109\cdot 353 + 309\cdot 353^{2} + 83\cdot 353^{3} + 79\cdot 353^{4} + 280\cdot 353^{5} + 241\cdot 353^{6} + 248\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 319 + 267\cdot 353 + 69\cdot 353^{2} + 172\cdot 353^{3} + 150\cdot 353^{4} + 222\cdot 353^{5} + 145\cdot 353^{6} + 259\cdot 353^{7} +O\left(353^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(3,6)(4,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,8)(4,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,5,8,4)(2,6,7,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(3,6)(4,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(3,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,6,7,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,2,5,6,8,7,4,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,5,2,8,6,4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
