Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 237\cdot 251 + 142\cdot 251^{2} + 15\cdot 251^{3} + 89\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 171\cdot 251 + 30\cdot 251^{2} + 109\cdot 251^{3} + 211\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 + 35\cdot 251 + 195\cdot 251^{2} + 102\cdot 251^{3} + 159\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 191 + 209\cdot 251 + 64\cdot 251^{2} + 3\cdot 251^{3} + 63\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 214 + 99\cdot 251 + 68\cdot 251^{2} + 20\cdot 251^{3} + 230\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
