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 + 21\cdot 59 + 15\cdot 59^{2} + 55\cdot 59^{3} + 2\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 22 + \left(39 a + 13\right)\cdot 59 + \left(2 a + 1\right)\cdot 59^{2} + \left(5 a + 48\right)\cdot 59^{3} + \left(28 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 25 + \left(24 a + 50\right)\cdot 59 + \left(22 a + 9\right)\cdot 59^{2} + \left(36 a + 31\right)\cdot 59^{3} + \left(42 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 6\cdot 59 + 59^{2} + 6\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 41 + \left(19 a + 33\right)\cdot 59 + \left(56 a + 23\right)\cdot 59^{2} + \left(53 a + 50\right)\cdot 59^{3} + \left(30 a + 22\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 48 + \left(34 a + 51\right)\cdot 59 + \left(36 a + 7\right)\cdot 59^{2} + \left(22 a + 45\right)\cdot 59^{3} + \left(16 a + 12\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,6)$ |
$0$ |
| $9$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,5,6,2)$ |
$1$ |
| $12$ |
$6$ |
$(2,4,5)(3,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
