Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 5\cdot 37 + 8\cdot 37^{2} + 35\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 28\cdot 37 + 7\cdot 37^{2} + 16\cdot 37^{3} + 12\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 18\cdot 37 + 16\cdot 37^{2} + 32\cdot 37^{3} + 13\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 11\cdot 37 + 20\cdot 37^{2} + 35\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 15\cdot 37 + 27\cdot 37^{2} + 25\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 28\cdot 37 + 9\cdot 37^{2} + 17\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 28\cdot 37 + 35\cdot 37^{2} + 22\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 11\cdot 37 + 22\cdot 37^{2} + 36\cdot 37^{3} + 27\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
