Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 51 + 28\cdot 389 + 342\cdot 389^{2} + 331\cdot 389^{3} + 351\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 86 + 12\cdot 389 + 9\cdot 389^{2} + 254\cdot 389^{3} + 333\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 95 + 181\cdot 389 + 100\cdot 389^{2} + 333\cdot 389^{3} + 16\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 113 + 322\cdot 389 + 132\cdot 389^{2} + 305\cdot 389^{3} + 7\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 124 + 88\cdot 389 + 269\cdot 389^{2} + 55\cdot 389^{3} + 96\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 160 + 297\cdot 389 + 103\cdot 389^{2} + 264\cdot 389^{3} + 203\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 210 + 272\cdot 389 + 17\cdot 389^{2} + 42\cdot 389^{3} + 85\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 328 + 353\cdot 389 + 191\cdot 389^{2} + 358\cdot 389^{3} + 71\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,6)(4,7)(5,8)$ |
| $(1,8)(5,6)$ |
| $(1,5,8,6)(2,7,3,4)$ |
| $(1,8)(2,3)(4,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,3)(4,7)(5,6)$ | $-4$ |
| $2$ | $2$ | $(1,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(4,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,6)(2,7,3,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,5)(2,7,3,4)$ | $0$ |
| $4$ | $4$ | $(1,2,8,3)(4,6,7,5)$ | $0$ |
| $4$ | $8$ | $(1,2,6,4,8,3,5,7)$ | $0$ |
| $4$ | $8$ | $(1,7,6,2,8,4,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
