Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 182 + 402\cdot 431 + 162\cdot 431^{2} + 224\cdot 431^{3} + 405\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 213 + 401\cdot 431 + 181\cdot 431^{2} + 331\cdot 431^{3} + 287\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 252 + 404\cdot 431 + 138\cdot 431^{2} + 317\cdot 431^{3} + 276\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 261 + 387\cdot 431 + 14\cdot 431^{2} + 76\cdot 431^{3} + 77\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 387 + 127\cdot 431 + 363\cdot 431^{2} + 343\cdot 431^{3} + 245\cdot 431^{4} +O\left(431^{ 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.
