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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 40\cdot 53 + 36\cdot 53^{2} + 23\cdot 53^{3} + 44\cdot 53^{4} + 52\cdot 53^{5} + 20\cdot 53^{6} + 38\cdot 53^{7} + 51\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 7 + \left(5 a + 7\right)\cdot 53 + \left(11 a + 52\right)\cdot 53^{2} + \left(25 a + 44\right)\cdot 53^{3} + \left(19 a + 19\right)\cdot 53^{4} + \left(27 a + 34\right)\cdot 53^{5} + \left(9 a + 17\right)\cdot 53^{6} + \left(47 a + 29\right)\cdot 53^{7} + \left(37 a + 37\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 12\cdot 53 + 31\cdot 53^{2} + 32\cdot 53^{3} + 10\cdot 53^{4} + 32\cdot 53^{5} + 38\cdot 53^{6} + 40\cdot 53^{7} + 52\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 13 + \left(21 a + 25\right)\cdot 53 + 5 a\cdot 53^{2} + \left(21 a + 28\right)\cdot 53^{3} + \left(8 a + 52\right)\cdot 53^{4} + \left(16 a + 8\right)\cdot 53^{5} + \left(3 a + 28\right)\cdot 53^{6} + \left(2 a + 24\right)\cdot 53^{7} + \left(2 a + 40\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 44 + \left(31 a + 36\right)\cdot 53 + 47 a\cdot 53^{2} + \left(31 a + 1\right)\cdot 53^{3} + \left(44 a + 12\right)\cdot 53^{4} + \left(36 a + 12\right)\cdot 53^{5} + \left(49 a + 25\right)\cdot 53^{6} + \left(50 a + 29\right)\cdot 53^{7} + \left(50 a + 46\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 32 + \left(47 a + 37\right)\cdot 53 + \left(41 a + 37\right)\cdot 53^{2} + \left(27 a + 28\right)\cdot 53^{3} + \left(33 a + 19\right)\cdot 53^{4} + \left(25 a + 18\right)\cdot 53^{5} + \left(43 a + 28\right)\cdot 53^{6} + \left(5 a + 49\right)\cdot 53^{7} + \left(15 a + 35\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(3,6)$ |
| $(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)(5,6)$ |
$-1$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $6$ |
$2$ |
$(1,5)(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,3,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,3,5)(2,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,6,4,3,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
