Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 14.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 3\cdot 37 + 37^{2} + 11\cdot 37^{3} + 7\cdot 37^{4} + 34\cdot 37^{5} + 31\cdot 37^{6} + 5\cdot 37^{7} + 5\cdot 37^{8} + 11\cdot 37^{9} + 3\cdot 37^{10} + 2\cdot 37^{11} + 29\cdot 37^{12} + 9\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 9\cdot 37 + 36\cdot 37^{2} + 9\cdot 37^{3} + 17\cdot 37^{4} + 9\cdot 37^{5} + 13\cdot 37^{6} + 30\cdot 37^{7} + 24\cdot 37^{8} + 17\cdot 37^{9} + 21\cdot 37^{10} + 32\cdot 37^{11} + 9\cdot 37^{12} + 13\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 4\cdot 37 + 12\cdot 37^{2} + 33\cdot 37^{4} + 17\cdot 37^{6} + 11\cdot 37^{7} + 35\cdot 37^{8} + 13\cdot 37^{9} + 2\cdot 37^{10} + 29\cdot 37^{11} + 8\cdot 37^{12} + 19\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 6\cdot 37 + 12\cdot 37^{2} + 36\cdot 37^{3} + 16\cdot 37^{4} + 15\cdot 37^{5} + 15\cdot 37^{6} + 20\cdot 37^{7} + 25\cdot 37^{8} + 29\cdot 37^{9} + 32\cdot 37^{10} + 30\cdot 37^{11} + 5\cdot 37^{12} + 28\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 30\cdot 37 + 24\cdot 37^{2} + 20\cdot 37^{4} + 21\cdot 37^{5} + 21\cdot 37^{6} + 16\cdot 37^{7} + 11\cdot 37^{8} + 7\cdot 37^{9} + 4\cdot 37^{10} + 6\cdot 37^{11} + 31\cdot 37^{12} + 8\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 32\cdot 37 + 24\cdot 37^{2} + 36\cdot 37^{3} + 3\cdot 37^{4} + 36\cdot 37^{5} + 19\cdot 37^{6} + 25\cdot 37^{7} + 37^{8} + 23\cdot 37^{9} + 34\cdot 37^{10} + 7\cdot 37^{11} + 28\cdot 37^{12} + 17\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 32 + 27\cdot 37 + 27\cdot 37^{3} + 19\cdot 37^{4} + 27\cdot 37^{5} + 23\cdot 37^{6} + 6\cdot 37^{7} + 12\cdot 37^{8} + 19\cdot 37^{9} + 15\cdot 37^{10} + 4\cdot 37^{11} + 27\cdot 37^{12} + 23\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 33\cdot 37 + 35\cdot 37^{2} + 25\cdot 37^{3} + 29\cdot 37^{4} + 2\cdot 37^{5} + 5\cdot 37^{6} + 31\cdot 37^{7} + 31\cdot 37^{8} + 25\cdot 37^{9} + 33\cdot 37^{10} + 34\cdot 37^{11} + 7\cdot 37^{12} + 27\cdot 37^{13} +O\left(37^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,5,6,8,2,4,3)$ |
| $(1,6,8,3)(2,5,7,4)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,3,8,7,4,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
