Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 6\cdot 37 + 22\cdot 37^{2} + 11\cdot 37^{3} + 13\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 18\cdot 37 + 3\cdot 37^{2} + 27\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 12\cdot 37 + 18\cdot 37^{2} + 12\cdot 37^{3} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 15\cdot 37 + 15\cdot 37^{2} + 15\cdot 37^{3} + 13\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 13\cdot 37 + 5\cdot 37^{2} + 27\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 11\cdot 37 + 29\cdot 37^{2} + 15\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 6\cdot 37 + 36\cdot 37^{2} + 25\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 35 + 25\cdot 37 + 17\cdot 37^{2} + 12\cdot 37^{3} + 6\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,5)(2,4,6,7)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,5)(2,4,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
