Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 24\cdot 37 + 8\cdot 37^{2} + 28\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 13\cdot 37 + 22\cdot 37^{2} + 19\cdot 37^{3} + 15\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 6\cdot 37 + 14\cdot 37^{2} + 20\cdot 37^{3} + 7\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 21\cdot 37 + 37^{2} + 25\cdot 37^{3} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 15\cdot 37 + 35\cdot 37^{2} + 11\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 30\cdot 37 + 22\cdot 37^{2} + 16\cdot 37^{3} + 29\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 + 23\cdot 37 + 14\cdot 37^{2} + 17\cdot 37^{3} + 21\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 12\cdot 37 + 28\cdot 37^{2} + 8\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,4,7)(2,8,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
