Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17\cdot 37 + 6\cdot 37^{2} + 20\cdot 37^{3} + 13\cdot 37^{4} + 26\cdot 37^{5} + 24\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 37 + 20\cdot 37^{2} + 3\cdot 37^{3} + 33\cdot 37^{4} + 6\cdot 37^{5} + 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 21\cdot 37 + 13\cdot 37^{2} + 35\cdot 37^{3} + 30\cdot 37^{4} + 11\cdot 37^{5} + 9\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 27\cdot 37 + 28\cdot 37^{2} + 34\cdot 37^{3} + 2\cdot 37^{4} + 30\cdot 37^{5} + 16\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 26\cdot 37 + 8\cdot 37^{2} + 4\cdot 37^{3} + 28\cdot 37^{4} + 3\cdot 37^{5} + 36\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 4\cdot 37 + 5\cdot 37^{2} + 37^{3} + 7\cdot 37^{4} + 6\cdot 37^{5} + 10\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 34\cdot 37 + 33\cdot 37^{2} + 14\cdot 37^{3} + 33\cdot 37^{4} + 28\cdot 37^{5} + 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 + 15\cdot 37 + 31\cdot 37^{2} + 33\cdot 37^{3} + 35\cdot 37^{4} + 33\cdot 37^{5} + 10\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,7)(4,5,8,6)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,2,3,7)(4,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
