Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 34\cdot 149 + 145\cdot 149^{2} + 97\cdot 149^{3} + 142\cdot 149^{4} + 82\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 64\cdot 149 + 14\cdot 149^{2} + 18\cdot 149^{3} + 111\cdot 149^{4} + 131\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 56\cdot 149 + 133\cdot 149^{2} + 127\cdot 149^{3} + 56\cdot 149^{4} + 50\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 115\cdot 149 + 57\cdot 149^{2} + 51\cdot 149^{3} + 7\cdot 149^{4} + 79\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 67\cdot 149 + 111\cdot 149^{2} + 38\cdot 149^{3} + 69\cdot 149^{4} + 69\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 39\cdot 149 + 11\cdot 149^{2} + 136\cdot 149^{3} + 95\cdot 149^{4} + 66\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 86 + 41\cdot 149 + 45\cdot 149^{2} + 99\cdot 149^{3} + 48\cdot 149^{4} + 79\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 101 + 29\cdot 149 + 77\cdot 149^{2} + 26\cdot 149^{3} + 64\cdot 149^{4} + 36\cdot 149^{5} +O\left(149^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(4,6)(5,7)$ |
| $(1,8,5,7)(2,6,3,4)$ |
| $(1,5)(2,3)(4,6)(7,8)$ |
| $(1,6,7,2,5,4,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(4,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,4,3,6)$ | $0$ |
| $2$ | $8$ | $(1,6,7,2,5,4,8,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,8,6,5,3,7,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
