Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a + 28 + \left(42 a + 15\right)\cdot 59 + \left(3 a + 12\right)\cdot 59^{2} + \left(46 a + 13\right)\cdot 59^{3} + \left(13 a + 26\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 20 + \left(42 a + 7\right)\cdot 59 + \left(3 a + 40\right)\cdot 59^{2} + \left(46 a + 12\right)\cdot 59^{3} + \left(13 a + 4\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 a + 29 + \left(16 a + 56\right)\cdot 59 + \left(55 a + 32\right)\cdot 59^{2} + \left(12 a + 55\right)\cdot 59^{3} + \left(45 a + 52\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 a + 11 + \left(16 a + 45\right)\cdot 59 + \left(55 a + 25\right)\cdot 59^{2} + \left(12 a + 41\right)\cdot 59^{3} + \left(45 a + 44\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a + 21 + \left(16 a + 48\right)\cdot 59 + \left(55 a + 1\right)\cdot 59^{2} + \left(12 a + 55\right)\cdot 59^{3} + \left(45 a + 30\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 10 + \left(42 a + 4\right)\cdot 59 + \left(3 a + 5\right)\cdot 59^{2} + \left(46 a + 58\right)\cdot 59^{3} + \left(13 a + 17\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,4,5)$ |
| $(1,3)(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,6,2)(3,4,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$6$ |
$(1,4,2,3,6,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,5,6,3,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
