Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 41\cdot 137 + 101\cdot 137^{2} + 69\cdot 137^{3} + 38\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 79\cdot 137 + 34\cdot 137^{2} + 124\cdot 137^{3} + 54\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 32\cdot 137 + 113\cdot 137^{2} + 89\cdot 137^{3} + 86\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 + 73\cdot 137 + 36\cdot 137^{2} + 119\cdot 137^{3} + 94\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 115\cdot 137 + 119\cdot 137^{2} + 84\cdot 137^{3} + 39\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 91\cdot 137 + 95\cdot 137^{2} + 34\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 105 + 55\cdot 137 + 110\cdot 137^{2} + 104\cdot 137^{3} + 70\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 106 + 58\cdot 137 + 73\cdot 137^{2} + 57\cdot 137^{3} + 109\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,7)(2,6,3,4)$ |
| $(1,5)(2,3)(4,6)(7,8)$ |
| $(1,3,7,6,5,2,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-1$ |
| $1$ | $4$ | $(1,7,5,8)(2,4,3,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,8,5,7)(2,6,3,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,3,7,6,5,2,8,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,8,3,5,4,7,2)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,2,7,4,5,3,8,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,8,2,5,6,7,3)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
