Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 49 + \left(58 a + 37\right)\cdot 59 + 46\cdot 59^{2} + \left(13 a + 1\right)\cdot 59^{3} + \left(41 a + 45\right)\cdot 59^{4} + \left(3 a + 16\right)\cdot 59^{5} + 15\cdot 59^{6} + \left(a + 22\right)\cdot 59^{7} + \left(34 a + 52\right)\cdot 59^{8} + \left(51 a + 21\right)\cdot 59^{9} + \left(19 a + 2\right)\cdot 59^{10} + \left(40 a + 1\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 16\cdot 59 + 23\cdot 59^{2} + 56\cdot 59^{3} + 8\cdot 59^{4} + 50\cdot 59^{5} + 30\cdot 59^{6} + 39\cdot 59^{7} + 56\cdot 59^{8} + 54\cdot 59^{9} + 42\cdot 59^{10} + 3\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 47 + \left(40 a + 26\right)\cdot 59 + \left(32 a + 52\right)\cdot 59^{2} + \left(13 a + 42\right)\cdot 59^{3} + \left(52 a + 56\right)\cdot 59^{4} + \left(11 a + 26\right)\cdot 59^{5} + \left(4 a + 28\right)\cdot 59^{6} + \left(3 a + 32\right)\cdot 59^{7} + \left(21 a + 15\right)\cdot 59^{8} + \left(19 a + 45\right)\cdot 59^{9} + \left(25 a + 57\right)\cdot 59^{10} + \left(57 a + 29\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 10 + 17\cdot 59 + \left(58 a + 48\right)\cdot 59^{2} + \left(45 a + 13\right)\cdot 59^{3} + \left(17 a + 14\right)\cdot 59^{4} + \left(55 a + 38\right)\cdot 59^{5} + \left(58 a + 11\right)\cdot 59^{6} + \left(57 a + 23\right)\cdot 59^{7} + \left(24 a + 26\right)\cdot 59^{8} + \left(7 a + 39\right)\cdot 59^{9} + \left(39 a + 29\right)\cdot 59^{10} + \left(18 a + 21\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 41\cdot 59 + 20\cdot 59^{2} + 38\cdot 59^{3} + 15\cdot 59^{4} + 58\cdot 59^{5} + 10\cdot 59^{6} + 28\cdot 59^{7} + 51\cdot 59^{8} + 30\cdot 59^{9} + 39\cdot 59^{10} + 58\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 18 + \left(18 a + 37\right)\cdot 59 + \left(26 a + 44\right)\cdot 59^{2} + \left(45 a + 23\right)\cdot 59^{3} + \left(6 a + 36\right)\cdot 59^{4} + \left(47 a + 45\right)\cdot 59^{5} + \left(54 a + 20\right)\cdot 59^{6} + \left(55 a + 31\right)\cdot 59^{7} + \left(37 a + 33\right)\cdot 59^{8} + \left(39 a + 43\right)\cdot 59^{9} + \left(33 a + 4\right)\cdot 59^{10} + \left(a + 3\right)\cdot 59^{11} +O\left(59^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(3,5)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,3,2)$ | $1$ |
| $6$ | $4$ | $(1,3)(2,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,2,3,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
