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:
| $r_{ 1 }$ |
$=$ |
$ 23 a^{2} + 10 a + 36 + \left(18 a^{2} + 41 a + 55\right)\cdot 59 + \left(58 a^{2} + 37 a + 39\right)\cdot 59^{2} + \left(6 a^{2} + 30 a + 12\right)\cdot 59^{3} + \left(57 a^{2} + 36 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a^{2} + 56 a + 18 + \left(48 a^{2} + 19 a + 28\right)\cdot 59 + \left(6 a^{2} + 56 a + 36\right)\cdot 59^{2} + \left(34 a^{2} + 15 a + 54\right)\cdot 59^{3} + \left(11 a^{2} + 27 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a^{2} + 17 a + 58 + \left(30 a^{2} + 34 a + 15\right)\cdot 59 + \left(39 a^{2} + 38 a + 36\right)\cdot 59^{2} + \left(29 a^{2} + 50 a + 9\right)\cdot 59^{3} + \left(4 a^{2} + 36 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a^{2} + 16 + \left(31 a^{2} + 58 a + 27\right)\cdot 59 + \left(39 a^{2} + 27 a + 36\right)\cdot 59^{2} + \left(29 a^{2} + 31 a + 50\right)\cdot 59^{3} + \left(13 a^{2} + 28 a + 13\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 a^{2} + 20 a + 39 + \left(51 a^{2} + 25 a + 39\right)\cdot 59 + \left(32 a^{2} + 12 a + 44\right)\cdot 59^{2} + \left(2 a^{2} + 7 a + 47\right)\cdot 59^{3} + \left(55 a^{2} + 53 a + 26\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a^{2} + 53 a + 23 + \left(17 a^{2} + 18 a + 20\right)\cdot 59 + \left(48 a^{2} + 10 a + 26\right)\cdot 59^{2} + \left(39 a^{2} + 10 a + 25\right)\cdot 59^{3} + \left(22 a^{2} + 9 a + 44\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 a^{2} + 42 a + 42 + \left(17 a^{2} + 13 a + 4\right)\cdot 59 + \left(19 a^{2} + 49 a + 19\right)\cdot 59^{2} + \left(22 a^{2} + 35 a + 15\right)\cdot 59^{3} + \left(51 a^{2} + 37 a + 34\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 a^{2} + 32 a + 20 + \left(10 a^{2} + 42 a + 29\right)\cdot 59 + \left(20 a^{2} + 41 a + 50\right)\cdot 59^{2} + \left(22 a^{2} + 36 a + 4\right)\cdot 59^{3} + \left(56 a^{2} + 44 a + 2\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 58 a^{2} + 6 a + 44 + \left(9 a^{2} + 41 a + 14\right)\cdot 59 + \left(30 a^{2} + 20 a + 5\right)\cdot 59^{2} + \left(48 a^{2} + 17 a + 15\right)\cdot 59^{3} + \left(22 a^{2} + 21 a + 25\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,5,6,8,7,4,3,2,9)$ |
| $(1,8,3)(2,5,7)(4,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,8,3)(2,5,7)(4,9,6)$ | $-1$ |
| $2$ | $9$ | $(1,5,6,8,7,4,3,2,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,6,7,3,9,5,8,4,2)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,7,9,8,2,6,3,5,4)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
