Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 84\cdot 239 + 43\cdot 239^{2} + 146\cdot 239^{3} + 146\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 49\cdot 239 + 145\cdot 239^{2} + 32\cdot 239^{3} + 31\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 100 + 92\cdot 239 + 45\cdot 239^{2} + 138\cdot 239^{3} + 67\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 109 + 80\cdot 239 + 23\cdot 239^{2} + 71\cdot 239^{3} + 120\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 115 + 212\cdot 239 + 39\cdot 239^{2} + 218\cdot 239^{3} + 74\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 124 + 200\cdot 239 + 17\cdot 239^{2} + 151\cdot 239^{3} + 127\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 188 + 135\cdot 239 + 30\cdot 239^{2} + 156\cdot 239^{3} + 12\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 202 + 100\cdot 239 + 132\cdot 239^{2} + 42\cdot 239^{3} + 136\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
