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:
| $r_{ 1 }$ |
$=$ |
$ 32 + 47\cdot 59 + 17\cdot 59^{2} + 39\cdot 59^{3} + 51\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 14 + \left(49 a + 16\right)\cdot 59 + \left(12 a + 26\right)\cdot 59^{2} + \left(39 a + 51\right)\cdot 59^{3} + \left(24 a + 55\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 11\cdot 59 + 43\cdot 59^{2} + 47\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 49 + \left(9 a + 30\right)\cdot 59 + \left(46 a + 48\right)\cdot 59^{2} + \left(19 a + 18\right)\cdot 59^{3} + \left(34 a + 41\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 55 + \left(16 a + 44\right)\cdot 59 + \left(2 a + 27\right)\cdot 59^{2} + \left(46 a + 17\right)\cdot 59^{3} + \left(45 a + 33\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a + 32 + \left(42 a + 25\right)\cdot 59 + \left(56 a + 13\right)\cdot 59^{2} + \left(12 a + 2\right)\cdot 59^{3} + \left(13 a + 33\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $4$ | $3$ | $(1,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,5,4,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
