Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 37 + 55\cdot 431 + 94\cdot 431^{2} + 109\cdot 431^{3} + 96\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 140 + 24\cdot 431 + 359\cdot 431^{2} + 374\cdot 431^{3} + 327\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 346 + 385\cdot 431 + 355\cdot 431^{2} + 262\cdot 431^{3} + 296\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 370 + 373\cdot 431 + 388\cdot 431^{2} + 363\cdot 431^{3} + 269\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 402 + 22\cdot 431 + 95\cdot 431^{2} + 182\cdot 431^{3} + 302\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
