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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 38 + \left(2 a + 3\right)\cdot 53 + \left(24 a + 13\right)\cdot 53^{2} + \left(35 a + 5\right)\cdot 53^{3} + \left(45 a + 13\right)\cdot 53^{4} + \left(43 a + 33\right)\cdot 53^{5} + \left(38 a + 19\right)\cdot 53^{6} + \left(34 a + 45\right)\cdot 53^{7} + \left(52 a + 36\right)\cdot 53^{8} + \left(34 a + 46\right)\cdot 53^{9} + \left(33 a + 47\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 37 + \left(50 a + 52\right)\cdot 53 + 28 a\cdot 53^{2} + \left(17 a + 17\right)\cdot 53^{3} + \left(7 a + 1\right)\cdot 53^{4} + \left(9 a + 4\right)\cdot 53^{5} + \left(14 a + 25\right)\cdot 53^{6} + \left(18 a + 39\right)\cdot 53^{7} + \left(18 a + 28\right)\cdot 53^{9} + \left(19 a + 41\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 37\cdot 53 + 7\cdot 53^{2} + 41\cdot 53^{3} + 3\cdot 53^{4} + 4\cdot 53^{5} + 40\cdot 53^{6} + 17\cdot 53^{7} + 14\cdot 53^{8} + 36\cdot 53^{9} + 10\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 28 + \left(43 a + 15\right)\cdot 53 + \left(49 a + 48\right)\cdot 53^{2} + \left(6 a + 35\right)\cdot 53^{3} + \left(42 a + 43\right)\cdot 53^{4} + \left(16 a + 15\right)\cdot 53^{5} + \left(40 a + 21\right)\cdot 53^{6} + \left(28 a + 45\right)\cdot 53^{7} + \left(8 a + 20\right)\cdot 53^{8} + \left(40 a + 43\right)\cdot 53^{9} + \left(22 a + 44\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 49\cdot 53 + 43\cdot 53^{2} + 45\cdot 53^{3} + 50\cdot 53^{4} + 7\cdot 53^{5} + 46\cdot 53^{6} + 49\cdot 53^{7} + 6\cdot 53^{8} + 21\cdot 53^{9} + 24\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 46 + \left(9 a + 52\right)\cdot 53 + \left(3 a + 44\right)\cdot 53^{2} + \left(46 a + 13\right)\cdot 53^{3} + \left(10 a + 46\right)\cdot 53^{4} + \left(36 a + 40\right)\cdot 53^{5} + \left(12 a + 6\right)\cdot 53^{6} + \left(24 a + 14\right)\cdot 53^{7} + \left(44 a + 26\right)\cdot 53^{8} + \left(12 a + 36\right)\cdot 53^{9} + \left(30 a + 42\right)\cdot 53^{10} +O\left(53^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(5,6)$ |
| $(3,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-3$ |
| $3$ |
$2$ |
$(3,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(3,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(4,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,6,5)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4,6,2)(3,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,2,3,6,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
