Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 16.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a^{2} + 21 a + 2 + \left(42 a^{2} + 25 a + 43\right)\cdot 61 + \left(5 a^{2} + 41 a + 2\right)\cdot 61^{2} + \left(25 a^{2} + 54 a + 18\right)\cdot 61^{3} + \left(30 a^{2} + 33 a + 2\right)\cdot 61^{4} + \left(56 a^{2} + 41 a + 5\right)\cdot 61^{5} + \left(60 a^{2} + 42 a + 37\right)\cdot 61^{6} + \left(22 a^{2} + 2 a + 37\right)\cdot 61^{7} + \left(6 a^{2} + 27 a + 38\right)\cdot 61^{8} + \left(33 a^{2} + 51 a + 47\right)\cdot 61^{9} + \left(34 a^{2} + 3 a + 53\right)\cdot 61^{10} + \left(28 a^{2} + 39 a + 47\right)\cdot 61^{11} + \left(2 a^{2} + 26 a + 59\right)\cdot 61^{12} + \left(a^{2} + 29 a + 53\right)\cdot 61^{13} + \left(21 a^{2} + 54 a + 29\right)\cdot 61^{14} + \left(4 a^{2} + 29 a + 33\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 14\cdot 61 + 13\cdot 61^{2} + 12\cdot 61^{3} + 13\cdot 61^{4} + 16\cdot 61^{5} + 48\cdot 61^{6} + 40\cdot 61^{7} + 52\cdot 61^{8} + 58\cdot 61^{9} + 4\cdot 61^{10} + 40\cdot 61^{11} + 42\cdot 61^{12} + 60\cdot 61^{13} + 2\cdot 61^{14} + 27\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a^{2} + 2 a + 28 + \left(43 a^{2} + 49 a + 52\right)\cdot 61 + \left(43 a^{2} + 44 a + 44\right)\cdot 61^{2} + \left(59 a^{2} + 9 a + 11\right)\cdot 61^{3} + \left(58 a^{2} + 11 a + 8\right)\cdot 61^{4} + \left(50 a^{2} + 55 a + 49\right)\cdot 61^{5} + \left(47 a^{2} + 52 a + 2\right)\cdot 61^{6} + \left(35 a^{2} + 51 a + 13\right)\cdot 61^{7} + \left(19 a^{2} + 38 a + 1\right)\cdot 61^{8} + \left(47 a^{2} + 27 a + 43\right)\cdot 61^{9} + \left(55 a^{2} + 13 a + 1\right)\cdot 61^{10} + \left(29 a^{2} + 41 a + 42\right)\cdot 61^{11} + \left(42 a^{2} + 48 a + 7\right)\cdot 61^{12} + \left(7 a^{2} + 53 a + 7\right)\cdot 61^{13} + \left(24 a + 28\right)\cdot 61^{14} + \left(25 a^{2} + a + 42\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a^{2} + 42 a + 14 + \left(a^{2} + 23 a + 36\right)\cdot 61 + \left(38 a^{2} + 3 a + 31\right)\cdot 61^{2} + \left(34 a^{2} + 16 a + 1\right)\cdot 61^{3} + \left(28 a^{2} + 38 a + 14\right)\cdot 61^{4} + \left(55 a^{2} + 13 a\right)\cdot 61^{5} + \left(47 a^{2} + 10 a + 17\right)\cdot 61^{6} + \left(12 a^{2} + 49 a + 10\right)\cdot 61^{7} + \left(13 a^{2} + 11 a + 50\right)\cdot 61^{8} + \left(14 a^{2} + 37 a + 40\right)\cdot 61^{9} + \left(21 a^{2} + 9 a + 52\right)\cdot 61^{10} + \left(a^{2} + 2 a + 42\right)\cdot 61^{11} + \left(40 a^{2} + 22 a + 31\right)\cdot 61^{12} + \left(6 a^{2} + 24 a + 39\right)\cdot 61^{13} + \left(40 a^{2} + 31 a + 37\right)\cdot 61^{14} + \left(20 a^{2} + 32 a + 48\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a^{2} + 40 a + 59 + \left(18 a^{2} + 35 a + 17\right)\cdot 61 + \left(55 a^{2} + 19 a + 58\right)\cdot 61^{2} + \left(35 a^{2} + 6 a + 42\right)\cdot 61^{3} + \left(30 a^{2} + 27 a + 58\right)\cdot 61^{4} + \left(4 a^{2} + 19 a + 55\right)\cdot 61^{5} + \left(18 a + 23\right)\cdot 61^{6} + \left(38 a^{2} + 58 a + 23\right)\cdot 61^{7} + \left(54 a^{2} + 33 a + 22\right)\cdot 61^{8} + \left(27 a^{2} + 9 a + 13\right)\cdot 61^{9} + \left(26 a^{2} + 57 a + 7\right)\cdot 61^{10} + \left(32 a^{2} + 21 a + 13\right)\cdot 61^{11} + \left(58 a^{2} + 34 a + 1\right)\cdot 61^{12} + \left(59 a^{2} + 31 a + 7\right)\cdot 61^{13} + \left(39 a^{2} + 6 a + 31\right)\cdot 61^{14} + \left(56 a^{2} + 31 a + 27\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 46\cdot 61 + 47\cdot 61^{2} + 48\cdot 61^{3} + 47\cdot 61^{4} + 44\cdot 61^{5} + 12\cdot 61^{6} + 20\cdot 61^{7} + 8\cdot 61^{8} + 2\cdot 61^{9} + 56\cdot 61^{10} + 20\cdot 61^{11} + 18\cdot 61^{12} + 58\cdot 61^{14} + 33\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 59 a + 33 + \left(17 a^{2} + 11 a + 8\right)\cdot 61 + \left(17 a^{2} + 16 a + 16\right)\cdot 61^{2} + \left(a^{2} + 51 a + 49\right)\cdot 61^{3} + \left(2 a^{2} + 49 a + 52\right)\cdot 61^{4} + \left(10 a^{2} + 5 a + 11\right)\cdot 61^{5} + \left(13 a^{2} + 8 a + 58\right)\cdot 61^{6} + \left(25 a^{2} + 9 a + 47\right)\cdot 61^{7} + \left(41 a^{2} + 22 a + 59\right)\cdot 61^{8} + \left(13 a^{2} + 33 a + 17\right)\cdot 61^{9} + \left(5 a^{2} + 47 a + 59\right)\cdot 61^{10} + \left(31 a^{2} + 19 a + 18\right)\cdot 61^{11} + \left(18 a^{2} + 12 a + 53\right)\cdot 61^{12} + \left(53 a^{2} + 7 a + 53\right)\cdot 61^{13} + \left(60 a^{2} + 36 a + 32\right)\cdot 61^{14} + \left(35 a^{2} + 59 a + 18\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 31 a^{2} + 19 a + 47 + \left(59 a^{2} + 37 a + 24\right)\cdot 61 + \left(22 a^{2} + 57 a + 29\right)\cdot 61^{2} + \left(26 a^{2} + 44 a + 59\right)\cdot 61^{3} + \left(32 a^{2} + 22 a + 46\right)\cdot 61^{4} + \left(5 a^{2} + 47 a + 60\right)\cdot 61^{5} + \left(13 a^{2} + 50 a + 43\right)\cdot 61^{6} + \left(48 a^{2} + 11 a + 50\right)\cdot 61^{7} + \left(47 a^{2} + 49 a + 10\right)\cdot 61^{8} + \left(46 a^{2} + 23 a + 20\right)\cdot 61^{9} + \left(39 a^{2} + 51 a + 8\right)\cdot 61^{10} + \left(59 a^{2} + 58 a + 18\right)\cdot 61^{11} + \left(20 a^{2} + 38 a + 29\right)\cdot 61^{12} + \left(54 a^{2} + 36 a + 21\right)\cdot 61^{13} + \left(20 a^{2} + 29 a + 23\right)\cdot 61^{14} + \left(40 a^{2} + 28 a + 12\right)\cdot 61^{15} +O\left(61^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,4)(7,8)$ |
| $(1,2,3,4)(5,6,7,8)$ |
| $(1,2,7,4)(3,8,5,6)$ |
| $(3,8)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $6$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $6$ | $2$ | $(1,5)(4,8)$ | $0$ |
| $12$ | $2$ | $(3,4)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $2$ |
| $32$ | $3$ | $(1,2,3)(5,6,7)$ | $1$ |
| $12$ | $4$ | $(1,7,5,3)(2,8,6,4)$ | $0$ |
| $24$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $24$ | $4$ | $(1,8,3,2)(4,7,6,5)$ | $0$ |
| $24$ | $4$ | $(1,5)(3,8,7,4)$ | $0$ |
| $32$ | $6$ | $(1,8,6,5,4,2)(3,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
