Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 116\cdot 137 + 13\cdot 137^{2} + 115\cdot 137^{3} + 95\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 68\cdot 137 + 10\cdot 137^{2} + 132\cdot 137^{3} + 39\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 63\cdot 137 + 53\cdot 137^{2} + 26\cdot 137^{3} + 61\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 37\cdot 137 + 83\cdot 137^{2} + 134\cdot 137^{3} + 86\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 126\cdot 137 + 79\cdot 137^{2} + 14\cdot 137^{3} + 31\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 91 + 105\cdot 137 + 126\cdot 137^{2} + 117\cdot 137^{3} + 85\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 113 + 4\cdot 137 + 121\cdot 137^{2} + 6\cdot 137^{3} + 70\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 26\cdot 137 + 59\cdot 137^{2} + 77\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,7)(3,4,5,8)$ |
| $(1,3)(2,8)(4,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,7)(3,4,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
