Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 401 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 150 + 146\cdot 401 + 320\cdot 401^{2} + 69\cdot 401^{3} + 339\cdot 401^{4} +O\left(401^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 191 + 4\cdot 401 + 236\cdot 401^{2} + 140\cdot 401^{3} + 244\cdot 401^{4} +O\left(401^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 242 + 126\cdot 401 + 319\cdot 401^{2} + 324\cdot 401^{3} + 153\cdot 401^{4} +O\left(401^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 296 + 372\cdot 401 + 118\cdot 401^{2} + 274\cdot 401^{3} + 66\cdot 401^{4} +O\left(401^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 325 + 151\cdot 401 + 208\cdot 401^{2} + 393\cdot 401^{3} + 398\cdot 401^{4} +O\left(401^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
