Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 32\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 27\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 2\cdot 61 + 22\cdot 61^{2} + 4\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 29\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 17\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 8\cdot 61 + 32\cdot 61^{2} + 12\cdot 61^{3} + 6\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 48\cdot 61 + 14\cdot 61^{2} + 50\cdot 61^{3} + 24\cdot 61^{4} +O\left(61^{ 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.
