Basic invariants
Defining polynomial
| $f(x)$ | $=$ | $ x^{4} - x^{3} - 2250 x^{2} + 12377 x + 116621 $. |
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 9\cdot 37 + 24\cdot 37^{2} + 15\cdot 37^{3} + 37^{4} + 20\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 20\cdot 37 + 17\cdot 37^{2} + 5\cdot 37^{3} + 10\cdot 37^{4} + 34\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 2\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{4} + 8\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 4\cdot 37 + 24\cdot 37^{2} + 15\cdot 37^{3} + 6\cdot 37^{4} + 11\cdot 37^{5} +O\left(37^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,3)(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,2,3,4)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
