Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 8\cdot 59 + 32\cdot 59^{2} + 46\cdot 59^{3} + 39\cdot 59^{4} + 8\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 16\cdot 59 + 9\cdot 59^{2} + 42\cdot 59^{3} + 4\cdot 59^{4} + 38\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 42\cdot 59 + 49\cdot 59^{2} + 16\cdot 59^{3} + 54\cdot 59^{4} + 20\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 50\cdot 59 + 26\cdot 59^{2} + 12\cdot 59^{3} + 19\cdot 59^{4} + 50\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3,4,2)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,3,4,2)$ |
$\zeta_{4}$ |
$-\zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,4,3)$ |
$-\zeta_{4}$ |
$\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
