Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 225\cdot 389 + 388\cdot 389^{2} + 271\cdot 389^{3} + 72\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 110 + 295\cdot 389 + 149\cdot 389^{2} + 111\cdot 389^{3} + 251\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 125 + 380\cdot 389 + 22\cdot 389^{2} + 183\cdot 389^{3} + 122\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 179 + 327\cdot 389 + 215\cdot 389^{2} + 366\cdot 389^{3} + 87\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 211 + 61\cdot 389 + 173\cdot 389^{2} + 22\cdot 389^{3} + 301\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 265 + 8\cdot 389 + 366\cdot 389^{2} + 205\cdot 389^{3} + 266\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 280 + 93\cdot 389 + 239\cdot 389^{2} + 277\cdot 389^{3} + 137\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 366 + 163\cdot 389 + 117\cdot 389^{3} + 316\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
