Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 171\cdot 457 + 240\cdot 457^{2} + 61\cdot 457^{3} + 205\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 102 + 353\cdot 457 + 118\cdot 457^{2} + 178\cdot 457^{3} + 93\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 149 + 343\cdot 457 + 127\cdot 457^{2} + 182\cdot 457^{3} + 368\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 237 + 251\cdot 457 + 79\cdot 457^{2} + 429\cdot 457^{3} + 235\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 382 + 251\cdot 457 + 347\cdot 457^{2} + 62\cdot 457^{3} + 11\cdot 457^{4} +O\left(457^{ 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.
