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:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 22 + \left(12 a + 33\right)\cdot 59 + \left(23 a + 29\right)\cdot 59^{2} + \left(28 a + 20\right)\cdot 59^{3} + \left(39 a + 38\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 50\cdot 59 + 50\cdot 59^{2} + 20\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 27\cdot 59 + 15\cdot 59^{2} + 58\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 17 + \left(46 a + 51\right)\cdot 59 + \left(35 a + 39\right)\cdot 59^{2} + \left(30 a + 25\right)\cdot 59^{3} + \left(19 a + 49\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 26 + \left(5 a + 14\right)\cdot 59 + \left(21 a + 42\right)\cdot 59^{2} + 12 a\cdot 59^{3} + \left(47 a + 48\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 46 + \left(53 a + 58\right)\cdot 59 + \left(37 a + 57\right)\cdot 59^{2} + \left(46 a + 50\right)\cdot 59^{3} + \left(11 a + 23\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,4)$ |
$2$ |
| $9$ |
$2$ |
$(3,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,3,6,4,2)$ |
$0$ |
| $12$ |
$6$ |
$(2,5,6)(3,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
