Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 a^{2} + 39 a + 52 + \left(7 a^{2} + 23 a + 36\right)\cdot 61 + \left(51 a^{2} + 15 a + 32\right)\cdot 61^{2} + \left(43 a^{2} + 44 a + 2\right)\cdot 61^{3} + \left(40 a^{2} + 27 a + 20\right)\cdot 61^{4} + \left(3 a^{2} + 47 a + 28\right)\cdot 61^{5} + \left(4 a^{2} + 35 a + 6\right)\cdot 61^{6} + \left(39 a^{2} + 56 a + 7\right)\cdot 61^{7} + \left(26 a^{2} + 53 a + 43\right)\cdot 61^{8} + \left(8 a^{2} + 16 a + 12\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 54 a + 21 + \left(9 a^{2} + a + 45\right)\cdot 61 + \left(17 a^{2} + 4 a + 36\right)\cdot 61^{2} + \left(59 a^{2} + 26 a + 13\right)\cdot 61^{3} + \left(40 a^{2} + 54 a + 21\right)\cdot 61^{4} + \left(48 a^{2} + 42 a + 55\right)\cdot 61^{5} + \left(12 a^{2} + 44 a + 26\right)\cdot 61^{6} + \left(26 a^{2} + 45 a + 28\right)\cdot 61^{7} + \left(59 a^{2} + 43 a + 33\right)\cdot 61^{8} + \left(25 a^{2} + 35 a + 33\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a^{2} + 52 a + 12 + \left(a^{2} + 42 a + 23\right)\cdot 61 + \left(51 a^{2} + 5 a + 14\right)\cdot 61^{2} + \left(2 a^{2} + 58 a + 8\right)\cdot 61^{3} + \left(24 a^{2} + 17 a + 8\right)\cdot 61^{4} + \left(32 a^{2} + 7 a + 52\right)\cdot 61^{5} + \left(18 a^{2} + 5 a + 20\right)\cdot 61^{6} + \left(a^{2} + 55 a + 17\right)\cdot 61^{7} + \left(15 a^{2} + 14 a + 13\right)\cdot 61^{8} + \left(58 a^{2} + 28 a + 25\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 31\cdot 61 + 41\cdot 61^{2} + 3\cdot 61^{3} + 49\cdot 61^{4} + 16\cdot 61^{5} + 30\cdot 61^{6} + 5\cdot 61^{7} + 38\cdot 61^{8} + 48\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 + 55\cdot 61 + 27\cdot 61^{2} + 17\cdot 61^{3} + 26\cdot 61^{4} + 48\cdot 61^{5} + 35\cdot 61^{6} + 28\cdot 61^{7} + 57\cdot 61^{8} + 43\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 a^{2} + 29 a + \left(44 a^{2} + 35 a + 48\right)\cdot 61 + \left(53 a^{2} + 41 a + 44\right)\cdot 61^{2} + \left(18 a^{2} + 51 a + 48\right)\cdot 61^{3} + \left(40 a^{2} + 39 a + 58\right)\cdot 61^{4} + \left(8 a^{2} + 31 a + 10\right)\cdot 61^{5} + \left(44 a^{2} + 41 a + 51\right)\cdot 61^{6} + \left(56 a^{2} + 19 a + 48\right)\cdot 61^{7} + \left(35 a^{2} + 24 a + 45\right)\cdot 61^{8} + \left(26 a^{2} + 8 a + 56\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 a^{2} + 36 a + 39 + \left(4 a^{2} + 60 a + 40\right)\cdot 61 + \left(13 a^{2} + 22 a + 40\right)\cdot 61^{2} + \left(55 a^{2} + 45 a + 8\right)\cdot 61^{3} + \left(49 a^{2} + 12 a + 27\right)\cdot 61^{4} + \left(7 a^{2} + 59\right)\cdot 61^{5} + \left(21 a^{2} + 16 a + 32\right)\cdot 61^{6} + \left(2 a^{2} + 29 a + 42\right)\cdot 61^{7} + \left(43 a^{2} + 14 a + 1\right)\cdot 61^{8} + \left(14 a^{2} + 15 a + 5\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 47 a^{2} + 34 a + 19 + \left(54 a^{2} + 18 a + 49\right)\cdot 61 + \left(57 a^{2} + 32 a + 5\right)\cdot 61^{2} + \left(2 a^{2} + 18 a + 29\right)\cdot 61^{3} + \left(48 a^{2} + 30 a + 18\right)\cdot 61^{4} + \left(20 a^{2} + 53 a + 18\right)\cdot 61^{5} + \left(21 a^{2} + 39 a + 54\right)\cdot 61^{6} + \left(57 a^{2} + 37 a + 34\right)\cdot 61^{7} + \left(2 a^{2} + 31 a + 58\right)\cdot 61^{8} + \left(49 a^{2} + 17 a + 22\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 52 + 35\cdot 61 + 60\cdot 61^{2} + 50\cdot 61^{3} + 14\cdot 61^{4} + 15\cdot 61^{5} + 46\cdot 61^{6} + 30\cdot 61^{7} + 13\cdot 61^{8} + 56\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5)(2,9)(4,6)$ |
| $(1,2,6)(4,9,5)$ |
| $(1,4,7)(2,5,3)(6,9,8)$ |
| $(1,2,6)(3,8,7)(4,5,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(3,4)(5,8)(7,9)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,8,7)(4,5,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,7,8)(4,9,5)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,4,7)(2,5,3)(6,9,8)$ | $0$ |
| $6$ | $3$ | $(1,9,8)(2,4,7)(3,6,5)$ | $0$ |
| $6$ | $3$ | $(1,5,3)(2,9,8)(4,7,6)$ | $0$ |
| $6$ | $3$ | $(1,2,6)(4,9,5)$ | $0$ |
| $9$ | $6$ | $(1,6,2)(3,9,8,4,7,5)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,2,6)(3,5,7,4,8,9)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
