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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 22 + \left(48 a + 34\right)\cdot 53 + \left(38 a + 39\right)\cdot 53^{2} + \left(49 a + 1\right)\cdot 53^{3} + \left(2 a + 42\right)\cdot 53^{4} + \left(3 a + 50\right)\cdot 53^{5} + 14 a\cdot 53^{6} + \left(8 a + 1\right)\cdot 53^{7} + \left(33 a + 17\right)\cdot 53^{8} + \left(38 a + 37\right)\cdot 53^{9} + \left(52 a + 46\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 21\cdot 53 + 25\cdot 53^{2} + 4\cdot 53^{3} + 53^{4} + 32\cdot 53^{5} + 27\cdot 53^{6} + 25\cdot 53^{7} + 5\cdot 53^{8} + 44\cdot 53^{9} + 15\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 27 + \left(17 a + 26\right)\cdot 53 + \left(17 a + 50\right)\cdot 53^{2} + \left(37 a + 41\right)\cdot 53^{3} + \left(a + 52\right)\cdot 53^{4} + \left(42 a + 40\right)\cdot 53^{5} + \left(36 a + 6\right)\cdot 53^{6} + \left(15 a + 21\right)\cdot 53^{7} + \left(12 a + 7\right)\cdot 53^{8} + \left(51 a + 6\right)\cdot 53^{9} + \left(30 a + 10\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 43 + \left(35 a + 37\right)\cdot 53 + \left(35 a + 49\right)\cdot 53^{2} + \left(15 a + 14\right)\cdot 53^{3} + \left(51 a + 22\right)\cdot 53^{4} + \left(10 a + 48\right)\cdot 53^{5} + \left(16 a + 5\right)\cdot 53^{6} + \left(37 a + 47\right)\cdot 53^{7} + \left(40 a + 40\right)\cdot 53^{8} + \left(a + 39\right)\cdot 53^{9} + \left(22 a + 29\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a + 42 + \left(4 a + 9\right)\cdot 53 + \left(14 a + 41\right)\cdot 53^{2} + \left(3 a + 2\right)\cdot 53^{3} + \left(50 a + 4\right)\cdot 53^{4} + \left(49 a + 7\right)\cdot 53^{5} + \left(38 a + 1\right)\cdot 53^{6} + \left(44 a + 20\right)\cdot 53^{7} + \left(19 a + 35\right)\cdot 53^{8} + \left(14 a + 52\right)\cdot 53^{9} + 6\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 28\cdot 53 + 5\cdot 53^{2} + 40\cdot 53^{3} + 36\cdot 53^{4} + 32\cdot 53^{5} + 10\cdot 53^{6} + 44\cdot 53^{7} + 52\cdot 53^{8} + 31\cdot 53^{9} + 49\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)(4,6,5)$ |
| $(1,3)(4,5)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,5)$ | $1$ |
| $6$ | $4$ | $(1,6,4,2)(3,5)$ | $-1$ |
| $8$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
