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