Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 268\cdot 379 + 13\cdot 379^{2} + 239\cdot 379^{3} + 191\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 190\cdot 379 + 22\cdot 379^{2} + 238\cdot 379^{3} + 73\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 330\cdot 379 + 86\cdot 379^{2} + 27\cdot 379^{3} + 22\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 106\cdot 379 + 260\cdot 379^{2} + 271\cdot 379^{3} + 9\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 224 + 46\cdot 379 + 321\cdot 379^{2} + 217\cdot 379^{3} + 219\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 238 + 156\cdot 379 + 243\cdot 379^{2} + 66\cdot 379^{3} + 271\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 259 + 205\cdot 379 + 314\cdot 379^{2} + 48\cdot 379^{3} + 261\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 278 + 212\cdot 379 + 253\cdot 379^{2} + 27\cdot 379^{3} + 88\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5,8,6,7,2)$ |
| $(1,6)(2,7)(3,8)(4,5)$ |
| $(2,5)(3,6)$ |
| $(1,4,8,7)(2,6,5,3)$ |
| $(1,8)(2,5)(3,6)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $-4$ |
| $2$ | $2$ | $(2,5)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,7)(2,3,5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,8,7)(2,6,5,3)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,7,5,4)$ | $0$ |
| $4$ | $8$ | $(1,3,4,5,8,6,7,2)$ | $0$ |
| $4$ | $8$ | $(1,6,7,5,8,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
