Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 a + 29 + \left(14 a + 44\right)\cdot 53 + \left(3 a + 37\right)\cdot 53^{2} + 14\cdot 53^{3} + \left(34 a + 18\right)\cdot 53^{4} + \left(30 a + 34\right)\cdot 53^{5} + \left(32 a + 1\right)\cdot 53^{6} + \left(19 a + 30\right)\cdot 53^{7} + \left(41 a + 11\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 47\cdot 53 + 39\cdot 53^{2} + 20\cdot 53^{3} + 23\cdot 53^{4} + 3\cdot 53^{5} + 44\cdot 53^{6} + 46\cdot 53^{7} + 6\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 33 + \left(14 a + 15\right)\cdot 53 + \left(3 a + 16\right)\cdot 53^{2} + 41\cdot 53^{3} + \left(34 a + 4\right)\cdot 53^{4} + \left(30 a + 36\right)\cdot 53^{5} + \left(32 a + 4\right)\cdot 53^{6} + \left(19 a + 30\right)\cdot 53^{7} + \left(41 a + 1\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 24 + \left(38 a + 8\right)\cdot 53 + \left(49 a + 15\right)\cdot 53^{2} + \left(52 a + 38\right)\cdot 53^{3} + \left(18 a + 34\right)\cdot 53^{4} + \left(22 a + 18\right)\cdot 53^{5} + \left(20 a + 51\right)\cdot 53^{6} + \left(33 a + 22\right)\cdot 53^{7} + \left(11 a + 41\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 5\cdot 53 + 13\cdot 53^{2} + 32\cdot 53^{3} + 29\cdot 53^{4} + 49\cdot 53^{5} + 8\cdot 53^{6} + 6\cdot 53^{7} + 46\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 20 + \left(38 a + 37\right)\cdot 53 + \left(49 a + 36\right)\cdot 53^{2} + \left(52 a + 11\right)\cdot 53^{3} + \left(18 a + 48\right)\cdot 53^{4} + \left(22 a + 16\right)\cdot 53^{5} + \left(20 a + 48\right)\cdot 53^{6} + \left(33 a + 22\right)\cdot 53^{7} + \left(11 a + 51\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,4,6)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4)(2,3,5,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
