Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ |
$=$ |
$ 12 + 43\cdot 59 + 3\cdot 59^{2} + 23\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
$r_{ 2 }$ |
$=$ |
$ 14 + 10\cdot 59 + 22\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
$r_{ 3 }$ |
$=$ |
$ 43 + 10\cdot 59 + 27\cdot 59^{2} + 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
$r_{ 4 }$ |
$=$ |
$ 50 + 53\cdot 59 + 5\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
Cycle notation |
$(1,4,2,3)$ |
$(1,2)(3,4)$ |
Character values on conjugacy classes
Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.