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