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:
| $r_{ 1 }$ |
$=$ |
$ 8 a^{2} + 18 a + 54 + \left(10 a^{2} + 37 a\right)\cdot 59 + \left(42 a^{2} + 14 a + 34\right)\cdot 59^{2} + \left(10 a^{2} + 16 a + 33\right)\cdot 59^{3} + \left(10 a^{2} + 5 a + 57\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 44 a + 4 + \left(53 a^{2} + 4 a + 25\right)\cdot 59 + \left(45 a^{2} + 3 a + 58\right)\cdot 59^{2} + \left(22 a^{2} + 2 a + 10\right)\cdot 59^{3} + \left(39 a^{2} + 36 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a^{2} + 38 a + 40 + \left(35 a^{2} + 17 a + 46\right)\cdot 59 + \left(31 a^{2} + 45 a + 50\right)\cdot 59^{2} + \left(35 a^{2} + 58 a + 47\right)\cdot 59^{3} + \left(8 a^{2} + 3 a + 31\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 39 a + 21 + \left(45 a^{2} + 12 a + 58\right)\cdot 59 + \left(33 a^{2} + 32 a + 5\right)\cdot 59^{2} + \left(27 a^{2} + 51 a + 11\right)\cdot 59^{3} + \left(17 a^{2} + 16 a + 3\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a^{2} + 50 a + 1 + \left(17 a^{2} + 58 a + 46\right)\cdot 59 + \left(20 a^{2} + 4 a + 32\right)\cdot 59^{2} + \left(34 a^{2} + 50 a + 4\right)\cdot 59^{3} + \left(32 a^{2} + 40 a + 33\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a^{2} + 30 a + 8 + \left(6 a^{2} + 41 a + 49\right)\cdot 59 + \left(7 a^{2} + 8 a + 47\right)\cdot 59^{2} + \left(48 a^{2} + 9 a + 30\right)\cdot 59^{3} + \left(17 a^{2} + 14 a + 3\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 a^{2} + 2 a + 7 + \left(3 a^{2} + 9 a + 19\right)\cdot 59 + \left(42 a^{2} + 12 a + 53\right)\cdot 59^{2} + \left(20 a^{2} + 50 a + 7\right)\cdot 59^{3} + \left(31 a^{2} + 36 a + 10\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 8 a^{2} + a + 4 + \left(7 a^{2} + 38 a + 29\right)\cdot 59 + \left(7 a^{2} + 24 a + 27\right)\cdot 59^{2} + \left(4 a^{2} + 2 a + 27\right)\cdot 59^{3} + \left(22 a^{2} + 44 a + 7\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 43 a^{2} + 14 a + 42 + \left(57 a^{2} + 16 a + 20\right)\cdot 59 + \left(5 a^{2} + 31 a + 43\right)\cdot 59^{2} + \left(32 a^{2} + 54 a + 2\right)\cdot 59^{3} + \left(56 a^{2} + 37 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,4)(2,8,9)(3,5,6)$ |
| $(1,2)(3,5)(4,8)(7,9)$ |
| $(1,3,9,7,5,2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)(4,8)(7,9)$ | $0$ |
| $2$ | $3$ | $(1,7,4)(2,8,9)(3,5,6)$ | $-1$ |
| $2$ | $9$ | $(1,3,9,7,5,2,4,6,8)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,9,5,4,8,3,7,2,6)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,5,8,7,6,9,4,3,2)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
