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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 7 + \left(41 a + 24\right)\cdot 59 + \left(14 a + 51\right)\cdot 59^{2} + \left(5 a + 55\right)\cdot 59^{3} + \left(2 a + 1\right)\cdot 59^{4} + 52\cdot 59^{5} + \left(21 a + 55\right)\cdot 59^{6} + \left(49 a + 54\right)\cdot 59^{7} + \left(56 a + 46\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 + 34\cdot 59 + 4\cdot 59^{2} + 20\cdot 59^{3} + 41\cdot 59^{4} + 17\cdot 59^{5} + 55\cdot 59^{6} + 26\cdot 59^{7} + 52\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 38 + \left(41 a + 7\right)\cdot 59 + \left(14 a + 34\right)\cdot 59^{2} + \left(5 a + 12\right)\cdot 59^{3} + \left(2 a + 1\right)\cdot 59^{4} + 9\cdot 59^{5} + \left(21 a + 41\right)\cdot 59^{6} + \left(49 a + 34\right)\cdot 59^{7} + \left(56 a + 4\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 52 + \left(17 a + 34\right)\cdot 59 + \left(44 a + 7\right)\cdot 59^{2} + \left(53 a + 3\right)\cdot 59^{3} + \left(56 a + 57\right)\cdot 59^{4} + \left(58 a + 6\right)\cdot 59^{5} + \left(37 a + 3\right)\cdot 59^{6} + \left(9 a + 4\right)\cdot 59^{7} + \left(2 a + 12\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 24\cdot 59 + 54\cdot 59^{2} + 38\cdot 59^{3} + 17\cdot 59^{4} + 41\cdot 59^{5} + 3\cdot 59^{6} + 32\cdot 59^{7} + 6\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 a + 21 + \left(17 a + 51\right)\cdot 59 + \left(44 a + 24\right)\cdot 59^{2} + \left(53 a + 46\right)\cdot 59^{3} + \left(56 a + 57\right)\cdot 59^{4} + \left(58 a + 49\right)\cdot 59^{5} + \left(37 a + 17\right)\cdot 59^{6} + \left(9 a + 24\right)\cdot 59^{7} + \left(2 a + 54\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $1$ |
| $8$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
