Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 20 + \left(29 a + 35\right)\cdot 53 + \left(44 a + 39\right)\cdot 53^{2} + \left(23 a + 51\right)\cdot 53^{3} + \left(7 a + 47\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 23\cdot 53 + 25\cdot 53^{2} + 23\cdot 53^{3} + 20\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 4 + \left(35 a + 33\right)\cdot 53 + \left(4 a + 26\right)\cdot 53^{2} + \left(39 a + 28\right)\cdot 53^{3} + \left(27 a + 37\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 51 + \left(17 a + 43\right)\cdot 53 + \left(48 a + 9\right)\cdot 53^{2} + \left(13 a + 21\right)\cdot 53^{3} + \left(25 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 27 + \left(23 a + 31\right)\cdot 53 + \left(8 a + 29\right)\cdot 53^{2} + \left(29 a + 49\right)\cdot 53^{3} + 45 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 44\cdot 53 + 27\cdot 53^{2} + 37\cdot 53^{3} + 48\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2,6,3,4)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$2$ |
| $15$ |
$2$ |
$(2,6)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $30$ |
$4$ |
$(2,4,6,3)$ |
$0$ |
| $24$ |
$5$ |
$(1,6,4,2,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,5,2,6,3,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
