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