Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 24 + \left(21 a + 35\right)\cdot 53 + \left(2 a + 40\right)\cdot 53^{2} + \left(2 a + 50\right)\cdot 53^{3} + \left(30 a + 32\right)\cdot 53^{4} + \left(39 a + 3\right)\cdot 53^{5} + \left(28 a + 16\right)\cdot 53^{6} + \left(5 a + 51\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 30 + \left(21 a + 51\right)\cdot 53 + \left(2 a + 23\right)\cdot 53^{2} + \left(2 a + 49\right)\cdot 53^{3} + \left(30 a + 7\right)\cdot 53^{4} + \left(39 a + 27\right)\cdot 53^{5} + \left(28 a + 14\right)\cdot 53^{6} + \left(5 a + 8\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 51\cdot 53 + 36\cdot 53^{2} + 12\cdot 53^{3} + 6\cdot 53^{4} + 42\cdot 53^{5} + 10\cdot 53^{6} + 8\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 29 + \left(31 a + 17\right)\cdot 53 + \left(50 a + 12\right)\cdot 53^{2} + \left(50 a + 2\right)\cdot 53^{3} + \left(22 a + 20\right)\cdot 53^{4} + \left(13 a + 49\right)\cdot 53^{5} + \left(24 a + 36\right)\cdot 53^{6} + \left(47 a + 1\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 23 + \left(31 a + 1\right)\cdot 53 + \left(50 a + 29\right)\cdot 53^{2} + \left(50 a + 3\right)\cdot 53^{3} + \left(22 a + 45\right)\cdot 53^{4} + \left(13 a + 25\right)\cdot 53^{5} + \left(24 a + 38\right)\cdot 53^{6} + \left(47 a + 44\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 53 + 16\cdot 53^{2} + 40\cdot 53^{3} + 46\cdot 53^{4} + 10\cdot 53^{5} + 42\cdot 53^{6} + 44\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(2,3)(5,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(2,3)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(2,3,5,6)$ |
$1$ |
| $6$ |
$4$ |
$(1,4)(2,3,5,6)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,5,4,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
