Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 138\cdot 151 + 113\cdot 151^{2} + 53\cdot 151^{3} + 32\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 69\cdot 151 + 73\cdot 151^{2} + 134\cdot 151^{3} + 121\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 97\cdot 151 + 136\cdot 151^{2} + 93\cdot 151^{3} + 121\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 2\cdot 151 + 68\cdot 151^{2} + 78\cdot 151^{3} + 132\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 + 20\cdot 151 + 90\cdot 151^{2} + 60\cdot 151^{3} + 70\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 113 + 105\cdot 151 + 4\cdot 151^{2} + 87\cdot 151^{3} + 108\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 117 + 81\cdot 151 + 68\cdot 151^{2} + 80\cdot 151^{3} + 141\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 143 + 88\cdot 151 + 48\cdot 151^{2} + 15\cdot 151^{3} + 26\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,8,5,6)(2,4,3,7)$ |
| $(1,5)(2,3)(4,7)(6,8)$ |
| $(1,2,8,4,5,3,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(4,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,2,8,4,5,3,6,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,6,2,5,7,8,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
