Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 43\cdot 151^{2} + 90\cdot 151^{3} + 130\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 + 69\cdot 151 + 91\cdot 151^{2} + 95\cdot 151^{3} + 58\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 80\cdot 151 + 120\cdot 151^{2} + 8\cdot 151^{3} + 70\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 + 129\cdot 151 + 96\cdot 151^{2} + 8\cdot 151^{3} + 93\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 + 21\cdot 151 + 54\cdot 151^{2} + 142\cdot 151^{3} + 57\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 70\cdot 151 + 30\cdot 151^{2} + 142\cdot 151^{3} + 80\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 106 + 81\cdot 151 + 59\cdot 151^{2} + 55\cdot 151^{3} + 92\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 + 150\cdot 151 + 107\cdot 151^{2} + 60\cdot 151^{3} + 20\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
