Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 373 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 71 + 45\cdot 373 + 337\cdot 373^{2} + 59\cdot 373^{3} + 310\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 105 + 171\cdot 373 + 232\cdot 373^{2} + 184\cdot 373^{3} + 211\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 218 + 9\cdot 373 + 315\cdot 373^{2} + 81\cdot 373^{3} + 245\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 363 + 53\cdot 373 + 34\cdot 373^{2} + 53\cdot 373^{3} + 141\cdot 373^{4} +O\left(373^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 364 + 92\cdot 373 + 200\cdot 373^{2} + 366\cdot 373^{3} + 210\cdot 373^{4} +O\left(373^{ 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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
