Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 + 283\cdot 431 + 194\cdot 431^{2} + 263\cdot 431^{3} + 86\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 237 + 167\cdot 431 + 161\cdot 431^{2} + 102\cdot 431^{3} + 76\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 305 + 397\cdot 431 + 274\cdot 431^{2} + 383\cdot 431^{3} + 337\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 349 + 335\cdot 431 + 238\cdot 431^{2} + 243\cdot 431^{3} + 178\cdot 431^{4} +O\left(431^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 361 + 108\cdot 431 + 423\cdot 431^{2} + 299\cdot 431^{3} + 182\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,2)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
