Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 4 + \left(12 a + 29\right)\cdot 59 + \left(48 a + 29\right)\cdot 59^{2} + \left(18 a + 22\right)\cdot 59^{3} + \left(13 a + 18\right)\cdot 59^{4} + \left(22 a + 37\right)\cdot 59^{5} + \left(23 a + 44\right)\cdot 59^{6} + \left(31 a + 18\right)\cdot 59^{7} + \left(21 a + 41\right)\cdot 59^{8} + \left(58 a + 16\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a + 14 + \left(46 a + 31\right)\cdot 59 + \left(10 a + 6\right)\cdot 59^{2} + \left(40 a + 52\right)\cdot 59^{3} + \left(45 a + 12\right)\cdot 59^{4} + \left(36 a + 46\right)\cdot 59^{5} + \left(35 a + 45\right)\cdot 59^{6} + \left(27 a + 26\right)\cdot 59^{7} + \left(37 a + 31\right)\cdot 59^{8} + 53\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 26 + \left(2 a + 16\right)\cdot 59 + \left(32 a + 35\right)\cdot 59^{2} + \left(20 a + 53\right)\cdot 59^{3} + \left(21 a + 12\right)\cdot 59^{4} + \left(35 a + 55\right)\cdot 59^{5} + 48 a\cdot 59^{6} + \left(46 a + 28\right)\cdot 59^{7} + \left(55 a + 46\right)\cdot 59^{8} + \left(49 a + 35\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 22\cdot 59 + 5\cdot 59^{2} + 22\cdot 59^{3} + 11\cdot 59^{4} + 33\cdot 59^{5} + 17\cdot 59^{6} + 53\cdot 59^{7} + 19\cdot 59^{8} + 5\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 8\cdot 59 + 35\cdot 59^{2} + 43\cdot 59^{3} + 48\cdot 59^{4} + 53\cdot 59^{5} + 53\cdot 59^{6} + 23\cdot 59^{7} + 41\cdot 59^{8} + 35\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a + 35 + \left(56 a + 9\right)\cdot 59 + \left(26 a + 6\right)\cdot 59^{2} + \left(38 a + 42\right)\cdot 59^{3} + \left(37 a + 13\right)\cdot 59^{4} + \left(23 a + 10\right)\cdot 59^{5} + \left(10 a + 14\right)\cdot 59^{6} + \left(12 a + 26\right)\cdot 59^{7} + \left(3 a + 55\right)\cdot 59^{8} + \left(9 a + 29\right)\cdot 59^{9} +O\left(59^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(5,6)$ |
| $(1,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(4,5)$ | $-1$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,4)(3,5)$ | $1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,4)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,5,3,4)$ | $1$ |
| $8$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
