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^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 22 + \left(3 a + 30\right)\cdot 53 + \left(26 a + 1\right)\cdot 53^{2} + \left(13 a + 12\right)\cdot 53^{3} + \left(35 a + 13\right)\cdot 53^{4} + \left(11 a + 22\right)\cdot 53^{5} + \left(28 a + 47\right)\cdot 53^{6} + \left(20 a + 5\right)\cdot 53^{7} + \left(44 a + 39\right)\cdot 53^{8} + \left(10 a + 26\right)\cdot 53^{9} + \left(42 a + 38\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 a + 38 + \left(41 a + 46\right)\cdot 53 + \left(35 a + 47\right)\cdot 53^{2} + \left(27 a + 35\right)\cdot 53^{3} + \left(46 a + 23\right)\cdot 53^{4} + \left(17 a + 4\right)\cdot 53^{5} + \left(15 a + 43\right)\cdot 53^{6} + \left(11 a + 41\right)\cdot 53^{7} + \left(33 a + 42\right)\cdot 53^{8} + \left(15 a + 26\right)\cdot 53^{9} + \left(33 a + 29\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 26 + \left(11 a + 5\right)\cdot 53 + \left(17 a + 43\right)\cdot 53^{2} + \left(25 a + 4\right)\cdot 53^{3} + \left(6 a + 23\right)\cdot 53^{4} + \left(35 a + 29\right)\cdot 53^{5} + \left(37 a + 33\right)\cdot 53^{6} + \left(41 a + 18\right)\cdot 53^{7} + \left(19 a + 5\right)\cdot 53^{8} + \left(37 a + 3\right)\cdot 53^{9} + \left(19 a + 41\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 33\cdot 53 + 9\cdot 53^{2} + 28\cdot 53^{3} + 30\cdot 53^{4} + 51\cdot 53^{5} + 5\cdot 53^{6} + 17\cdot 53^{7} + 46\cdot 53^{8} + 7\cdot 53^{9} + 19\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 31 + \left(49 a + 3\right)\cdot 53 + \left(26 a + 49\right)\cdot 53^{2} + \left(39 a + 39\right)\cdot 53^{3} + \left(17 a + 34\right)\cdot 53^{4} + \left(41 a + 33\right)\cdot 53^{5} + \left(24 a + 42\right)\cdot 53^{6} + \left(32 a + 6\right)\cdot 53^{7} + \left(8 a + 37\right)\cdot 53^{8} + \left(42 a + 25\right)\cdot 53^{9} + \left(10 a + 37\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 39\cdot 53 + 7\cdot 53^{2} + 38\cdot 53^{3} + 33\cdot 53^{4} + 17\cdot 53^{5} + 39\cdot 53^{6} + 15\cdot 53^{7} + 41\cdot 53^{8} + 15\cdot 53^{9} + 46\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2,4)(3,5,6)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,3,2)$ | $1$ |
| $6$ | $4$ | $(1,5,3,2)(4,6)$ | $-1$ |
| $8$ | $6$ | $(1,5,6,3,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
