Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 + 49\cdot 59 + 38\cdot 59^{2} + 30\cdot 59^{3} + 5\cdot 59^{4} + 38\cdot 59^{5} + 54\cdot 59^{6} + 46\cdot 59^{7} + 2\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 26 + \left(46 a + 38\right)\cdot 59 + \left(19 a + 9\right)\cdot 59^{2} + \left(17 a + 1\right)\cdot 59^{3} + \left(37 a + 24\right)\cdot 59^{4} + \left(19 a + 11\right)\cdot 59^{5} + \left(2 a + 18\right)\cdot 59^{6} + \left(51 a + 25\right)\cdot 59^{7} + \left(13 a + 13\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 39 + \left(36 a + 23\right)\cdot 59 + 37 a\cdot 59^{2} + \left(38 a + 28\right)\cdot 59^{3} + 3 a\cdot 59^{4} + \left(52 a + 45\right)\cdot 59^{5} + \left(43 a + 47\right)\cdot 59^{6} + \left(46 a + 18\right)\cdot 59^{7} + \left(38 a + 21\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 10 + \left(22 a + 30\right)\cdot 59 + \left(21 a + 1\right)\cdot 59^{2} + \left(20 a + 29\right)\cdot 59^{3} + \left(55 a + 24\right)\cdot 59^{4} + \left(6 a + 34\right)\cdot 59^{5} + \left(15 a + 39\right)\cdot 59^{6} + \left(12 a + 21\right)\cdot 59^{7} + \left(20 a + 13\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 31\cdot 59 + 25\cdot 59^{2} + 30\cdot 59^{3} + 19\cdot 59^{4} + 54\cdot 59^{5} + 15\cdot 59^{6} + 49\cdot 59^{7} + 31\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a + 48 + \left(12 a + 3\right)\cdot 59 + \left(39 a + 42\right)\cdot 59^{2} + \left(41 a + 57\right)\cdot 59^{3} + \left(21 a + 43\right)\cdot 59^{4} + \left(39 a + 52\right)\cdot 59^{5} + 56 a\cdot 59^{6} + \left(7 a + 15\right)\cdot 59^{7} + \left(45 a + 35\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,6)$ |
| $(3,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,3,4,6)$ | $1$ |
| $6$ | $4$ | $(1,4,5,2)(3,6)$ | $-1$ |
| $8$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
