Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 30\cdot 37 + 18\cdot 37^{2} + 35\cdot 37^{3} + 17\cdot 37^{4} + 26\cdot 37^{5} + 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 13\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{3} + 5\cdot 37^{4} + 22\cdot 37^{5} + 28\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 4\cdot 37 + 18\cdot 37^{2} + 34\cdot 37^{3} + 22\cdot 37^{4} + 32\cdot 37^{5} + 25\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 2\cdot 37 + 9\cdot 37^{2} + 35\cdot 37^{3} + 35\cdot 37^{5} + 21\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 34\cdot 37 + 27\cdot 37^{2} + 37^{3} + 36\cdot 37^{4} + 37^{5} + 15\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 32\cdot 37 + 18\cdot 37^{2} + 2\cdot 37^{3} + 14\cdot 37^{4} + 4\cdot 37^{5} + 11\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 23\cdot 37 + 28\cdot 37^{2} + 17\cdot 37^{3} + 31\cdot 37^{4} + 14\cdot 37^{5} + 8\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 34 + 6\cdot 37 + 18\cdot 37^{2} + 37^{3} + 19\cdot 37^{4} + 10\cdot 37^{5} + 35\cdot 37^{6} +O\left(37^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
