Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 25\cdot 37 + 26\cdot 37^{2} + 14\cdot 37^{3} + 31\cdot 37^{4} + 3\cdot 37^{5} + 5\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 4\cdot 37 + 3\cdot 37^{2} + 22\cdot 37^{3} + 33\cdot 37^{4} + 6\cdot 37^{5} + 35\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 8\cdot 37 + 24\cdot 37^{2} + 27\cdot 37^{3} + 33\cdot 37^{4} + 22\cdot 37^{5} + 14\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 17\cdot 37^{2} + 27\cdot 37^{3} + 24\cdot 37^{4} + 33\cdot 37^{5} + 17\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 36\cdot 37 + 19\cdot 37^{2} + 9\cdot 37^{3} + 12\cdot 37^{4} + 3\cdot 37^{5} + 19\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 + 28\cdot 37 + 12\cdot 37^{2} + 9\cdot 37^{3} + 3\cdot 37^{4} + 14\cdot 37^{5} + 22\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 32\cdot 37 + 33\cdot 37^{2} + 14\cdot 37^{3} + 3\cdot 37^{4} + 30\cdot 37^{5} + 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 34 + 11\cdot 37 + 10\cdot 37^{2} + 22\cdot 37^{3} + 5\cdot 37^{4} + 33\cdot 37^{5} + 31\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,5)(4,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
