Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 a^{2} + 39 a + 2 + \left(14 a^{2} + 23 a + 29\right)\cdot 59 + \left(25 a^{2} + 54 a + 5\right)\cdot 59^{2} + \left(21 a^{2} + 23 a + 32\right)\cdot 59^{3} + \left(32 a^{2} + 18 a + 9\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a^{2} + 25 a + 1 + \left(5 a^{2} + 46 a + 57\right)\cdot 59 + \left(26 a^{2} + 51 a + 27\right)\cdot 59^{2} + \left(25 a^{2} + 57 a + 45\right)\cdot 59^{3} + \left(52 a^{2} + 26 a + 56\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a^{2} + 27 a + 19 + \left(11 a^{2} + 31 a + 39\right)\cdot 59 + \left(2 a^{2} + 46 a + 46\right)\cdot 59^{2} + \left(36 a^{2} + 49 a + 21\right)\cdot 59^{3} + \left(19 a^{2} + 4 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a^{2} + 4 a + 26 + \left(14 a^{2} + 57 a + 10\right)\cdot 59 + \left(25 a^{2} + 47 a + 25\right)\cdot 59^{2} + \left(29 a^{2} + 24 a + 19\right)\cdot 59^{3} + \left(53 a^{2} + 28 a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a^{2} + 23 a + 27 + \left(18 a^{2} + 46 a + 21\right)\cdot 59 + \left(30 a^{2} + 41 a + 22\right)\cdot 59^{2} + \left(10 a^{2} + 30 a + 15\right)\cdot 59^{3} + \left(9 a^{2} + 31 a + 50\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a^{2} + 56 a + 30 + \left(26 a^{2} + 47 a + 8\right)\cdot 59 + \left(3 a^{2} + 21 a + 31\right)\cdot 59^{2} + \left(27 a^{2} + 4 a + 11\right)\cdot 59^{3} + \left(17 a^{2} + 9 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 30 a^{2} + 55 a + 41 + \left(44 a^{2} + 2 a + 10\right)\cdot 59 + \left(45 a^{2} + 25 a + 54\right)\cdot 59^{2} + \left(48 a^{2} + 49 a + 24\right)\cdot 59^{3} + \left(21 a^{2} + 17 a + 33\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 16 a^{2} + 28 a + 14 + \left(32 a^{2} + 29 a + 9\right)\cdot 59 + \left(31 a^{2} + 23 a + 46\right)\cdot 59^{2} + \left(52 a^{2} + 43 a + 17\right)\cdot 59^{3} + \left(44 a^{2} + 25 a + 51\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 11 a^{2} + 38 a + 17 + \left(9 a^{2} + 9 a + 50\right)\cdot 59 + \left(46 a^{2} + 41 a + 35\right)\cdot 59^{2} + \left(43 a^{2} + 10 a + 47\right)\cdot 59^{3} + \left(43 a^{2} + 14 a + 27\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,6,5)(2,9,7)(3,4,8)$ |
| $(1,5)(2,4)(3,9)(7,8)$ |
| $(1,7,4,6,2,8,5,9,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,4)(3,9)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,6,5)(2,9,7)(3,4,8)$ | $-1$ |
| $2$ | $9$ | $(1,7,4,6,2,8,5,9,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,4,2,5,3,7,6,8,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,2,3,6,9,4,5,7,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
