Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 118\cdot 241 + 117\cdot 241^{2} + 122\cdot 241^{3} + 121\cdot 241^{4} +O\left(241^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 36\cdot 241 + 87\cdot 241^{2} + 225\cdot 241^{3} + 196\cdot 241^{4} +O\left(241^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 139 + 146\cdot 241 + 17\cdot 241^{2} + 224\cdot 241^{3} + 23\cdot 241^{4} +O\left(241^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 154 + 71\cdot 241 + 21\cdot 241^{2} + 205\cdot 241^{3} + 198\cdot 241^{4} +O\left(241^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 169 + 109\cdot 241 + 238\cdot 241^{2} + 186\cdot 241^{3} + 181\cdot 241^{4} +O\left(241^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
