Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 60\cdot 137 + 117\cdot 137^{2} + 111\cdot 137^{3} + 42\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 13\cdot 137 + 60\cdot 137^{2} + 118\cdot 137^{3} + 89\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 74 + 9\cdot 137 + 52\cdot 137^{2} + 86\cdot 137^{3} + 25\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 131\cdot 137 + 16\cdot 137^{2} + 132\cdot 137^{3} + 40\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 80 + 59\cdot 137 + 27\cdot 137^{2} + 99\cdot 137^{3} + 74\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
