Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 a + 13 + \left(25 a + 8\right)\cdot 59 + \left(6 a + 44\right)\cdot 59^{2} + \left(24 a + 24\right)\cdot 59^{3} + \left(18 a + 5\right)\cdot 59^{4} + \left(41 a + 41\right)\cdot 59^{5} + \left(3 a + 11\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 58 + \left(33 a + 47\right)\cdot 59 + \left(52 a + 24\right)\cdot 59^{2} + \left(34 a + 42\right)\cdot 59^{3} + \left(40 a + 58\right)\cdot 59^{4} + \left(17 a + 4\right)\cdot 59^{5} + \left(55 a + 33\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 17\cdot 59^{2} + 47\cdot 59^{3} + 31\cdot 59^{4} + 32\cdot 59^{5} + 12\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 a + 47 + \left(24 a + 33\right)\cdot 59 + \left(42 a + 41\right)\cdot 59^{2} + \left(6 a + 23\right)\cdot 59^{3} + \left(20 a + 36\right)\cdot 59^{4} + \left(3 a + 21\right)\cdot 59^{5} + \left(23 a + 13\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 22 + \left(34 a + 24\right)\cdot 59 + 16 a\cdot 59^{2} + \left(52 a + 47\right)\cdot 59^{3} + \left(38 a + 49\right)\cdot 59^{4} + \left(55 a + 4\right)\cdot 59^{5} + \left(35 a + 33\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 + 2\cdot 59 + 49\cdot 59^{2} + 50\cdot 59^{3} + 53\cdot 59^{4} + 12\cdot 59^{5} + 14\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,3)(2,4)(5,6)$ |
| $(2,6)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,2)(3,4,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,4,6,5,2,3)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
