Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 18\cdot 37 + 6\cdot 37^{2} + 13\cdot 37^{3} + 11\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 33\cdot 37 + 15\cdot 37^{2} + 22\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 32\cdot 37 + 31\cdot 37^{2} + 28\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 27\cdot 37 + 19\cdot 37^{2} + 9\cdot 37^{3} + 31\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
