Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 373 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 + 261\cdot 373 + 255\cdot 373^{2} + 264\cdot 373^{3} + 334\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 103 + 265\cdot 373 + 244\cdot 373^{2} + 218\cdot 373^{3} + 143\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 116 + 47\cdot 373 + 130\cdot 373^{2} + 21\cdot 373^{3} + 289\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 260 + 312\cdot 373 + 26\cdot 373^{2} + 108\cdot 373^{3} + 282\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 290 + 16\cdot 373 + 346\cdot 373^{2} + 51\cdot 373^{3} + 290\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 320 + 366\cdot 373 + 179\cdot 373^{2} + 186\cdot 373^{3} + 76\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 338 + 141\cdot 373 + 257\cdot 373^{2} + 52\cdot 373^{3} + 295\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 367 + 79\cdot 373 + 51\cdot 373^{2} + 215\cdot 373^{3} + 153\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(4,8)(5,6)$ |
| $(1,7,2,3)(4,8,5,6)$ |
| $(4,5)(6,8)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,8)(2,6)(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-4$ |
| $2$ | $2$ | $(4,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,2,3)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,2,3)(4,6,5,8)$ | $0$ |
| $4$ | $4$ | $(1,6,2,8)(3,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,4,7,8,2,5,3,6)$ | $0$ |
| $4$ | $8$ | $(1,8,7,5,2,6,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
