Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 23\cdot 37 + 6\cdot 37^{2} + 8\cdot 37^{3} + 33\cdot 37^{4} + 25\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 4\cdot 37 + 35\cdot 37^{2} + 9\cdot 37^{3} + 37^{4} + 27\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 11\cdot 37 + 20\cdot 37^{2} + 5\cdot 37^{3} + 20\cdot 37^{4} + 19\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 37 + 25\cdot 37^{2} + 23\cdot 37^{3} + 17\cdot 37^{4} + 35\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 35\cdot 37 + 11\cdot 37^{2} + 13\cdot 37^{3} + 19\cdot 37^{4} + 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 25\cdot 37 + 16\cdot 37^{2} + 31\cdot 37^{3} + 16\cdot 37^{4} + 17\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 + 32\cdot 37 + 37^{2} + 27\cdot 37^{3} + 35\cdot 37^{4} + 9\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 13\cdot 37 + 30\cdot 37^{2} + 28\cdot 37^{3} + 3\cdot 37^{4} + 11\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,3)(4,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
