Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 5 + \left(36 a + 44\right)\cdot 59 + \left(31 a + 52\right)\cdot 59^{2} + \left(6 a + 21\right)\cdot 59^{3} + \left(35 a + 2\right)\cdot 59^{4} + \left(8 a + 46\right)\cdot 59^{5} + \left(52 a + 58\right)\cdot 59^{6} + \left(8 a + 56\right)\cdot 59^{7} + \left(6 a + 50\right)\cdot 59^{8} + \left(29 a + 40\right)\cdot 59^{9} + 20 a\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 52 + \left(5 a + 58\right)\cdot 59 + \left(52 a + 47\right)\cdot 59^{2} + \left(29 a + 28\right)\cdot 59^{3} + \left(4 a + 23\right)\cdot 59^{4} + \left(23 a + 34\right)\cdot 59^{5} + \left(35 a + 15\right)\cdot 59^{6} + \left(25 a + 20\right)\cdot 59^{7} + \left(49 a + 40\right)\cdot 59^{8} + \left(18 a + 41\right)\cdot 59^{9} + \left(40 a + 55\right)\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 22 + \left(22 a + 4\right)\cdot 59 + \left(27 a + 48\right)\cdot 59^{2} + \left(52 a + 55\right)\cdot 59^{3} + \left(23 a + 30\right)\cdot 59^{4} + \left(50 a + 19\right)\cdot 59^{5} + \left(6 a + 43\right)\cdot 59^{6} + \left(50 a + 13\right)\cdot 59^{7} + \left(52 a + 48\right)\cdot 59^{8} + \left(29 a + 4\right)\cdot 59^{9} + \left(38 a + 51\right)\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 37\cdot 59 + 15\cdot 59^{2} + 39\cdot 59^{3} + 22\cdot 59^{4} + 52\cdot 59^{5} + 11\cdot 59^{6} + 25\cdot 59^{7} + 5\cdot 59^{8} + 7\cdot 59^{9} + 37\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 47\cdot 59 + 35\cdot 59^{2} + 24\cdot 59^{3} + 40\cdot 59^{4} + 30\cdot 59^{5} + 19\cdot 59^{6} + 50\cdot 59^{7} + 26\cdot 59^{8} + 12\cdot 59^{9} + 14\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 13 + \left(53 a + 44\right)\cdot 59 + \left(6 a + 35\right)\cdot 59^{2} + \left(29 a + 6\right)\cdot 59^{3} + \left(54 a + 57\right)\cdot 59^{4} + \left(35 a + 52\right)\cdot 59^{5} + \left(23 a + 27\right)\cdot 59^{6} + \left(33 a + 10\right)\cdot 59^{7} + \left(9 a + 5\right)\cdot 59^{8} + \left(40 a + 11\right)\cdot 59^{9} + \left(18 a + 18\right)\cdot 59^{10} +O\left(59^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)$ |
| $(1,2)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,2,6)$ |
$1$ |
| $6$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,3,2,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
