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