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:
| $r_{ 1 }$ |
$=$ |
$ 20 + 47\cdot 59 + 56\cdot 59^{2} + 23\cdot 59^{3} + 23\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 a + 29 + \left(37 a + 23\right)\cdot 59 + \left(53 a + 51\right)\cdot 59^{2} + \left(18 a + 16\right)\cdot 59^{3} + \left(32 a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 26 + \left(21 a + 5\right)\cdot 59 + \left(5 a + 8\right)\cdot 59^{2} + \left(40 a + 41\right)\cdot 59^{3} + \left(26 a + 14\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 a + 47 + \left(2 a + 51\right)\cdot 59 + \left(14 a + 20\right)\cdot 59^{2} + \left(34 a + 39\right)\cdot 59^{3} + \left(11 a + 40\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 41 + \left(56 a + 1\right)\cdot 59 + \left(44 a + 32\right)\cdot 59^{2} + 24 a\cdot 59^{3} + \left(47 a + 18\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 47\cdot 59 + 7\cdot 59^{2} + 55\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$2$ |
| $6$ |
$2$ |
$(2,3)$ |
$0$ |
| $9$ |
$2$ |
$(2,3)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $4$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,4)(2,6,3,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,2,6,3,4)$ |
$-1$ |
| $12$ |
$6$ |
$(2,3)(4,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
