Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 37 + 102\cdot 151 + 25\cdot 151^{2} + 128\cdot 151^{3} + 9\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 + 136\cdot 151 + 17\cdot 151^{2} + 72\cdot 151^{3} + 28\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 87 + 144\cdot 151 + 67\cdot 151^{2} + 117\cdot 151^{3} + 76\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 97 + 144\cdot 151 + 96\cdot 151^{2} + 24\cdot 151^{3} + 45\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 86\cdot 151 + 65\cdot 151^{2} + 67\cdot 151^{3} + 5\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 106 + 86\cdot 151 + 5\cdot 151^{2} + 6\cdot 151^{3} + 69\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 131 + 38\cdot 151 + 149\cdot 151^{2} + 55\cdot 151^{3} + 41\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 132 + 14\cdot 151 + 24\cdot 151^{2} + 132\cdot 151^{3} + 25\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,4,3,2,7,8,6)$ |
| $(3,6)(5,7)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,8,2,4)(3,7,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(3,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,4,2,8)(3,7,6,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,2,4)(3,5,6,7)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,4)(3,7,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,4,3,2,7,8,6)$ | $0$ |
| $2$ | $8$ | $(1,3,8,5,2,6,4,7)$ | $0$ |
| $2$ | $8$ | $(1,3,4,7,2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,7,8,3,2,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
