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