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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 32 + \left(17 a + 9\right)\cdot 53 + \left(31 a + 6\right)\cdot 53^{2} + \left(22 a + 32\right)\cdot 53^{3} + \left(3 a + 6\right)\cdot 53^{4} + \left(45 a + 42\right)\cdot 53^{5} + \left(45 a + 28\right)\cdot 53^{6} + \left(19 a + 19\right)\cdot 53^{7} + 43 a\cdot 53^{8} + 12\cdot 53^{9} + \left(43 a + 4\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 51 + \left(23 a + 25\right)\cdot 53 + \left(37 a + 10\right)\cdot 53^{2} + \left(36 a + 50\right)\cdot 53^{3} + \left(15 a + 27\right)\cdot 53^{4} + \left(13 a + 42\right)\cdot 53^{5} + \left(23 a + 28\right)\cdot 53^{6} + \left(50 a + 38\right)\cdot 53^{7} + \left(37 a + 52\right)\cdot 53^{8} + \left(42 a + 38\right)\cdot 53^{9} + \left(2 a + 14\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 22 + \left(29 a + 6\right)\cdot 53 + \left(15 a + 31\right)\cdot 53^{2} + 16 a\cdot 53^{3} + \left(37 a + 1\right)\cdot 53^{4} + \left(39 a + 27\right)\cdot 53^{5} + \left(29 a + 2\right)\cdot 53^{6} + \left(2 a + 5\right)\cdot 53^{7} + \left(15 a + 48\right)\cdot 53^{8} + \left(10 a + 12\right)\cdot 53^{9} + \left(50 a + 36\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 50 + \left(35 a + 48\right)\cdot 53 + \left(21 a + 7\right)\cdot 53^{2} + \left(30 a + 38\right)\cdot 53^{3} + \left(49 a + 50\right)\cdot 53^{4} + \left(7 a + 6\right)\cdot 53^{5} + \left(7 a + 8\right)\cdot 53^{6} + 33 a\cdot 53^{7} + \left(9 a + 48\right)\cdot 53^{8} + \left(52 a + 24\right)\cdot 53^{9} + \left(9 a + 16\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 32\cdot 53 + 5\cdot 53^{2} + 37\cdot 53^{3} + 14\cdot 53^{4} + 12\cdot 53^{5} + 48\cdot 53^{6} + 39\cdot 53^{7} + 17\cdot 53^{8} + 50\cdot 53^{9} + 39\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 35\cdot 53 + 44\cdot 53^{2} + 5\cdot 53^{4} + 28\cdot 53^{5} + 42\cdot 53^{6} + 2\cdot 53^{7} + 45\cdot 53^{8} + 19\cdot 53^{9} + 47\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(3,6,4,5)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,6,4,5)$ | $1$ |
| $8$ | $6$ | $(1,3,6,2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
