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:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 46 + \left(5 a + 11\right)\cdot 53 + \left(20 a + 5\right)\cdot 53^{2} + 31 a\cdot 53^{3} + \left(30 a + 47\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 41 + \left(33 a + 5\right)\cdot 53 + \left(45 a + 47\right)\cdot 53^{2} + \left(38 a + 16\right)\cdot 53^{3} + \left(46 a + 44\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a + 9 + \left(47 a + 28\right)\cdot 53 + \left(32 a + 27\right)\cdot 53^{2} + \left(21 a + 52\right)\cdot 53^{3} + \left(22 a + 31\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 23 + \left(19 a + 11\right)\cdot 53 + \left(7 a + 37\right)\cdot 53^{2} + \left(14 a + 20\right)\cdot 53^{3} + \left(6 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 48\cdot 53 + 41\cdot 53^{2} + 15\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
