Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 22\cdot 37 + 28\cdot 37^{2} + 25\cdot 37^{3} + 37^{4} + 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 10\cdot 37 + 37^{2} + 14\cdot 37^{3} + 19\cdot 37^{4} + 33\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 35\cdot 37 + 36\cdot 37^{2} + 22\cdot 37^{3} + 25\cdot 37^{4} + 20\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 24\cdot 37 + 3\cdot 37^{2} + 14\cdot 37^{3} + 34\cdot 37^{4} + 10\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 25\cdot 37 + 33\cdot 37^{2} + 37^{3} + 18\cdot 37^{4} + 15\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 4\cdot 37 + 32\cdot 37^{2} + 22\cdot 37^{3} + 31\cdot 37^{4} + 8\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 + 29\cdot 37 + 4\cdot 37^{2} + 11\cdot 37^{3} + 12\cdot 37^{4} + 4\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 35 + 33\cdot 37 + 6\cdot 37^{2} + 35\cdot 37^{3} + 4\cdot 37^{4} + 16\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,8)(4,5)(6,7)$ |
| $(1,3,7,4)(2,5,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,4)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,4)(2,5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
