Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 117\cdot 137 + 114\cdot 137^{2} + 91\cdot 137^{3} + 39\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 + 23\cdot 137 + 86\cdot 137^{2} + 129\cdot 137^{3} + 112\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 110\cdot 137 + 95\cdot 137^{2} + 86\cdot 137^{3} + 7\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 63\cdot 137 + 115\cdot 137^{2} + 25\cdot 137^{3} + 18\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 98 + 126\cdot 137 + 49\cdot 137^{2} + 135\cdot 137^{3} + 131\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 119 + 3\cdot 137 + 40\cdot 137^{2} + 111\cdot 137^{3} + 40\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 130 + 119\cdot 137 + 84\cdot 137^{2} + 69\cdot 137^{3} + 71\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 135 + 119\cdot 137 + 97\cdot 137^{2} + 34\cdot 137^{3} + 125\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,7)(5,6)$ |
| $(1,3)(4,7)(5,6)$ |
| $(1,3,4,7)(2,6,8,5)$ |
| $(1,6)(2,3)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,7)(5,6)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,6)(2,3)(4,5)(7,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(4,7)(5,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,7)(2,6,8,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,7,8,4,6,3,2)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,3,5,4,2,7,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
