Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 163\cdot 379 + 133\cdot 379^{2} + 6\cdot 379^{3} + 130\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 95 + 108\cdot 379 + 294\cdot 379^{2} + 278\cdot 379^{3} + 334\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 142 + 238\cdot 379 + 161\cdot 379^{2} + 106\cdot 379^{3} + 231\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 157 + 123\cdot 379 + 314\cdot 379^{2} + 322\cdot 379^{3} + 296\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 188 + 272\cdot 379 + 11\cdot 379^{2} + 112\cdot 379^{3} + 211\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 252 + 173\cdot 379 + 171\cdot 379^{2} + 325\cdot 379^{3} + 275\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 309 + 153\cdot 379 + 176\cdot 379^{2} + 168\cdot 379^{3} + 181\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 371 + 282\cdot 379 + 252\cdot 379^{2} + 195\cdot 379^{3} + 233\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)$ |
| $(1,7)(2,4)(3,5)(6,8)$ |
| $(1,3,2,8)(4,5,7,6)$ |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,8)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $-4$ |
| $2$ | $2$ | $(1,2)(3,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,3)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,2,8)(4,5,7,6)$ | $0$ |
| $2$ | $4$ | $(1,8,2,3)(4,5,7,6)$ | $0$ |
| $4$ | $4$ | $(1,7,2,4)(3,5,8,6)$ | $0$ |
| $4$ | $8$ | $(1,7,8,5,2,4,3,6)$ | $0$ |
| $4$ | $8$ | $(1,6,8,7,2,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
