Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{3} + 3 x + 51 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 34 + \left(27 a^{2} + 45 a + 29\right)\cdot 53 + \left(34 a^{2} + 16 a + 51\right)\cdot 53^{2} + \left(36 a^{2} + 17 a + 1\right)\cdot 53^{3} + \left(a^{2} + 17 a + 47\right)\cdot 53^{4} + \left(34 a^{2} + 35 a + 37\right)\cdot 53^{5} + \left(51 a^{2} + 52 a + 49\right)\cdot 53^{6} + \left(25 a^{2} + 4 a + 48\right)\cdot 53^{7} + \left(42 a^{2} + 29 a + 5\right)\cdot 53^{8} + \left(51 a + 9\right)\cdot 53^{9} + \left(23 a^{2} + 6 a + 18\right)\cdot 53^{10} + \left(7 a^{2} + 32 a + 30\right)\cdot 53^{11} + \left(6 a^{2} + 36 a + 30\right)\cdot 53^{12} + \left(28 a^{2} + 22 a + 24\right)\cdot 53^{13} + \left(29 a^{2} + 35 a + 24\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a^{2} + 33 a + 36 + \left(15 a^{2} + 2 a + 5\right)\cdot 53 + \left(49 a^{2} + 15 a + 28\right)\cdot 53^{2} + \left(43 a + 36\right)\cdot 53^{3} + \left(41 a^{2} + 42 a + 19\right)\cdot 53^{4} + \left(36 a^{2} + 10 a + 43\right)\cdot 53^{5} + \left(12 a^{2} + 46 a + 24\right)\cdot 53^{6} + \left(22 a^{2} + 11 a + 41\right)\cdot 53^{7} + \left(5 a^{2} + 34 a + 37\right)\cdot 53^{8} + \left(49 a^{2} + 10 a + 52\right)\cdot 53^{9} + \left(45 a^{2} + 28 a + 10\right)\cdot 53^{10} + \left(42 a^{2} + 16 a + 48\right)\cdot 53^{11} + \left(32 a^{2} + 24 a + 30\right)\cdot 53^{12} + \left(8 a^{2} + 30 a + 38\right)\cdot 53^{13} + \left(44 a^{2} + 3 a\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 a^{2} + 31 a + 32 + \left(10 a^{2} + 4 a + 50\right)\cdot 53 + \left(22 a^{2} + 21 a + 26\right)\cdot 53^{2} + \left(15 a^{2} + 45 a + 12\right)\cdot 53^{3} + \left(10 a^{2} + 45 a + 11\right)\cdot 53^{4} + \left(35 a^{2} + 6 a + 40\right)\cdot 53^{5} + \left(41 a^{2} + 7 a + 29\right)\cdot 53^{6} + \left(4 a^{2} + 36 a + 6\right)\cdot 53^{7} + \left(5 a^{2} + 42 a + 37\right)\cdot 53^{8} + \left(3 a^{2} + 43 a + 13\right)\cdot 53^{9} + \left(37 a^{2} + 17 a + 46\right)\cdot 53^{10} + \left(2 a^{2} + 4 a + 20\right)\cdot 53^{11} + \left(14 a^{2} + 45 a + 46\right)\cdot 53^{12} + \left(16 a^{2} + 52 a\right)\cdot 53^{13} + \left(32 a^{2} + 13 a + 30\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 38\cdot 53 + 5\cdot 53^{2} + 29\cdot 53^{3} + 10\cdot 53^{4} + 22\cdot 53^{5} + 20\cdot 53^{6} + 26\cdot 53^{7} + 52\cdot 53^{8} + 3\cdot 53^{9} + 44\cdot 53^{10} + 19\cdot 53^{11} + 4\cdot 53^{12} + 28\cdot 53^{13} + 17\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 15\cdot 53 + 3\cdot 53^{2} + 25\cdot 53^{3} + 53^{4} + 41\cdot 53^{5} + 52\cdot 53^{6} + 49\cdot 53^{7} + 34\cdot 53^{8} + 42\cdot 53^{9} + 46\cdot 53^{10} + 12\cdot 53^{11} + 35\cdot 53^{12} + 38\cdot 53^{13} + 47\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a^{2} + 50 a + 3 + \left(20 a^{2} + 48 a + 12\right)\cdot 53 + \left(23 a^{2} + 29 a + 8\right)\cdot 53^{2} + \left(10 a^{2} + 26 a + 21\right)\cdot 53^{3} + \left(38 a + 23\right)\cdot 53^{4} + \left(43 a^{2} + 46 a + 24\right)\cdot 53^{5} + \left(42 a^{2} + 34 a + 26\right)\cdot 53^{6} + \left(14 a^{2} + 4 a + 42\right)\cdot 53^{7} + \left(5 a^{2} + 39 a + 42\right)\cdot 53^{8} + \left(7 a^{2} + 26 a + 8\right)\cdot 53^{9} + \left(43 a^{2} + a + 13\right)\cdot 53^{10} + \left(15 a^{2} + 2 a + 5\right)\cdot 53^{11} + \left(18 a^{2} + 37 a + 5\right)\cdot 53^{12} + \left(50 a^{2} + 45 a + 4\right)\cdot 53^{13} + \left(24 a^{2} + 47 a + 45\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 a^{2} + 16 a + 13 + \left(30 a^{2} + 2 a + 32\right)\cdot 53 + \left(31 a^{2} + 27 a + 24\right)\cdot 53^{2} + \left(16 a^{2} + 26 a + 33\right)\cdot 53^{3} + \left(a^{2} + 26 a + 25\right)\cdot 53^{4} + \left(17 a^{2} + 2 a + 25\right)\cdot 53^{5} + \left(2 a^{2} + 32 a + 51\right)\cdot 53^{6} + \left(9 a^{2} + 16 a + 30\right)\cdot 53^{7} + \left(19 a^{2} + 8 a + 17\right)\cdot 53^{8} + \left(42 a^{2} + 18 a + 26\right)\cdot 53^{9} + \left(39 a^{2} + 48 a + 6\right)\cdot 53^{10} + \left(30 a^{2} + 49 a + 35\right)\cdot 53^{11} + \left(3 a^{2} + 37 a + 28\right)\cdot 53^{12} + \left(51 a^{2} + 17 a + 5\right)\cdot 53^{13} + \left(16 a^{2} + 30 a + 29\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 a^{2} + 40 a + 25 + \left(a^{2} + a + 27\right)\cdot 53 + \left(51 a^{2} + 49 a + 10\right)\cdot 53^{2} + \left(25 a^{2} + 52 a + 52\right)\cdot 53^{3} + \left(51 a^{2} + 40 a + 19\right)\cdot 53^{4} + \left(45 a^{2} + 3 a + 30\right)\cdot 53^{5} + \left(7 a^{2} + 39 a + 9\right)\cdot 53^{6} + \left(29 a^{2} + 31 a + 18\right)\cdot 53^{7} + \left(28 a^{2} + 5 a + 36\right)\cdot 53^{8} + \left(3 a^{2} + 8 a + 1\right)\cdot 53^{9} + \left(23 a^{2} + 3 a + 26\right)\cdot 53^{10} + \left(6 a^{2} + a + 39\right)\cdot 53^{11} + \left(31 a^{2} + 31 a + 30\right)\cdot 53^{12} + \left(4 a^{2} + 42 a + 18\right)\cdot 53^{13} + \left(11 a^{2} + 27 a + 17\right)\cdot 53^{14} +O\left(53^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,2)(5,7,8,6)$ |
| $(1,3,4,7)(2,8,6,5)$ |
| $(1,7)(2,8)$ |
| $(1,2)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$8$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-8$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $6$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $6$ |
$2$ |
$(1,8)(3,6)$ |
$0$ |
| $12$ |
$2$ |
$(1,7)(2,8)$ |
$0$ |
| $12$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $32$ |
$3$ |
$(2,5,3)(4,6,7)$ |
$-1$ |
| $12$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $24$ |
$4$ |
$(1,3,4,2)(5,7,8,6)$ |
$0$ |
| $24$ |
$4$ |
$(1,7,5,6)(2,4,3,8)$ |
$0$ |
| $24$ |
$4$ |
$(1,7,8,2)(3,6)$ |
$0$ |
| $32$ |
$6$ |
$(1,8)(2,6,5,7,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
