Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 9\cdot 61 + 3\cdot 61^{2} + 36\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 17\cdot 61 + 46\cdot 61^{2} + 35\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 43\cdot 61 + 14\cdot 61^{2} + 25\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 51\cdot 61 + 57\cdot 61^{2} + 24\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
