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