Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 32\cdot 37 + 27\cdot 37^{2} + 11\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 35\cdot 37 + 27\cdot 37^{2} + 22\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 2\cdot 37 + 9\cdot 37^{2} + 9\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 31\cdot 37 + 15\cdot 37^{2} + 34\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 6\cdot 37 + 31\cdot 37^{2} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 10\cdot 37 + 14\cdot 37^{2} + 6\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 37 + 34\cdot 37^{2} + 17\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 26\cdot 37 + 24\cdot 37^{2} + 16\cdot 37^{3} + 11\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,7)(6,8)$ |
| $(1,3,6,4)(2,7,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,8)(3,4)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,3)(4,8)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,6,4)(2,7,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
