Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a^{2} + a + 47 + \left(56 a^{2} + 20 a + 8\right)\cdot 59 + \left(16 a^{2} + 24 a + 40\right)\cdot 59^{2} + \left(29 a^{2} + 52 a + 37\right)\cdot 59^{3} + \left(17 a^{2} + 56 a + 5\right)\cdot 59^{4} + \left(9 a^{2} + 49\right)\cdot 59^{5} + \left(21 a^{2} + 49 a + 15\right)\cdot 59^{6} + \left(39 a^{2} + 43 a + 39\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 2\cdot 59 + 35\cdot 59^{2} + 24\cdot 59^{4} + 48\cdot 59^{5} + 13\cdot 59^{6} + 3\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 58 a^{2} + 50 a + 58 + \left(46 a + 54\right)\cdot 59 + \left(16 a^{2} + 28 a + 24\right)\cdot 59^{2} + \left(21 a^{2} + 25 a + 38\right)\cdot 59^{3} + \left(55 a^{2} + 54 a + 29\right)\cdot 59^{4} + \left(38 a^{2} + 7 a + 14\right)\cdot 59^{5} + \left(33 a^{2} + 4\right)\cdot 59^{6} + \left(14 a^{2} + 8 a\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 29\cdot 59 + 52\cdot 59^{2} + 31\cdot 59^{3} + 8\cdot 59^{4} + 44\cdot 59^{6} + 38\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 a^{2} + 56 a + 36 + \left(26 a^{2} + 42 a + 48\right)\cdot 59 + \left(10 a^{2} + 27 a + 57\right)\cdot 59^{2} + \left(25 a^{2} + 43 a + 23\right)\cdot 59^{3} + \left(57 a^{2} + 18 a + 40\right)\cdot 59^{4} + \left(14 a^{2} + 15 a + 28\right)\cdot 59^{5} + \left(4 a^{2} + 17 a + 18\right)\cdot 59^{6} + \left(50 a^{2} + 42 a + 55\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 a^{2} + 47 a + 21 + \left(43 a^{2} + 7 a + 20\right)\cdot 59 + \left(53 a^{2} + 38 a + 52\right)\cdot 59^{2} + \left(16 a^{2} + 43 a + 23\right)\cdot 59^{3} + \left(57 a^{2} + 29 a + 16\right)\cdot 59^{4} + \left(9 a^{2} + 27 a + 16\right)\cdot 59^{5} + \left(57 a^{2} + 52 a + 43\right)\cdot 59^{6} + \left(32 a^{2} + 49 a + 41\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 51\cdot 59 + 46\cdot 59^{2} + 8\cdot 59^{3} + 7\cdot 59^{5} + 17\cdot 59^{6} + 25\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 19 a^{2} + 21 a + 46 + \left(14 a^{2} + 4 a + 20\right)\cdot 59 + \left(48 a^{2} + 51 a + 14\right)\cdot 59^{2} + \left(20 a^{2} + 48 a + 17\right)\cdot 59^{3} + \left(5 a^{2} + 33 a + 20\right)\cdot 59^{4} + \left(10 a^{2} + 23 a + 36\right)\cdot 59^{5} + \left(27 a^{2} + 6 a + 41\right)\cdot 59^{6} + \left(11 a^{2} + a + 48\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 15 a^{2} + 2 a + 28 + \left(35 a^{2} + 55 a + 57\right)\cdot 59 + \left(31 a^{2} + 6 a + 29\right)\cdot 59^{2} + \left(4 a^{2} + 22 a + 53\right)\cdot 59^{3} + \left(43 a^{2} + 42 a + 31\right)\cdot 59^{4} + \left(34 a^{2} + 42 a + 35\right)\cdot 59^{5} + \left(33 a^{2} + 51 a + 37\right)\cdot 59^{6} + \left(28 a^{2} + 31 a + 42\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,5,6)(7,9,8)$ |
| $(1,9,5)(3,6,8)$ |
| $(2,4,7)(3,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $3$ | $(1,5,9)(2,4,7)(3,6,8)$ | $-3 \zeta_{3} - 3$ |
| $1$ | $3$ | $(1,9,5)(2,7,4)(3,8,6)$ | $3 \zeta_{3}$ |
| $3$ | $3$ | $(1,3,2)(4,5,6)(7,9,8)$ | $0$ |
| $3$ | $3$ | $(1,2,3)(4,6,5)(7,8,9)$ | $0$ |
| $3$ | $3$ | $(1,9,5)(3,6,8)$ | $0$ |
| $3$ | $3$ | $(1,5,9)(3,8,6)$ | $0$ |
| $3$ | $3$ | $(1,8,2)(3,4,5)(6,7,9)$ | $0$ |
| $3$ | $3$ | $(1,2,8)(3,5,4)(6,9,7)$ | $0$ |
| $3$ | $3$ | $(1,7,8)(2,3,5)(4,6,9)$ | $0$ |
| $3$ | $3$ | $(1,8,7)(2,5,3)(4,9,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
