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 a + 7 + \left(46 a + 8\right)\cdot 59 + \left(32 a + 51\right)\cdot 59^{2} + \left(30 a + 48\right)\cdot 59^{3} + \left(18 a + 5\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 49 + \left(46 a + 29\right)\cdot 59 + \left(15 a + 58\right)\cdot 59^{2} + \left(32 a + 8\right)\cdot 59^{3} + \left(42 a + 50\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 16\cdot 59 + 29\cdot 59^{2} + 22\cdot 59^{3} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 21\cdot 59 + 31\cdot 59^{2} + 24\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a + 58 + \left(12 a + 7\right)\cdot 59 + \left(43 a + 28\right)\cdot 59^{2} + \left(26 a + 25\right)\cdot 59^{3} + \left(16 a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 27 + \left(12 a + 34\right)\cdot 59 + \left(26 a + 37\right)\cdot 59^{2} + \left(28 a + 46\right)\cdot 59^{3} + \left(40 a + 52\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(1,3)$ |
$0$ |
| $9$ |
$2$ |
$(1,3)(2,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,4,3,2)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,5,6,2)$ |
$1$ |
| $12$ |
$6$ |
$(1,3)(2,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
