Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 29 + \left(7 a + 24\right)\cdot 53 + \left(35 a + 2\right)\cdot 53^{2} + \left(22 a + 28\right)\cdot 53^{3} + \left(15 a + 2\right)\cdot 53^{4} + \left(47 a + 35\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 a + 37 + \left(12 a + 47\right)\cdot 53 + \left(22 a + 11\right)\cdot 53^{2} + \left(11 a + 32\right)\cdot 53^{3} + \left(52 a + 21\right)\cdot 53^{4} + \left(49 a + 29\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 37\cdot 53 + 47\cdot 53^{2} + 50\cdot 53^{3} + 49\cdot 53^{4} + 28\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 13 + \left(45 a + 7\right)\cdot 53 + \left(17 a + 29\right)\cdot 53^{2} + \left(30 a + 30\right)\cdot 53^{3} + \left(37 a + 41\right)\cdot 53^{4} + \left(5 a + 49\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 35\cdot 53 + 32\cdot 53^{2} + 14\cdot 53^{3} + 36\cdot 53^{4} + 50\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 34 + \left(40 a + 6\right)\cdot 53 + \left(30 a + 35\right)\cdot 53^{2} + \left(41 a + 2\right)\cdot 53^{3} + 7\cdot 53^{4} + \left(3 a + 18\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(4,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $-1$ |
| $20$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $12$ | $5$ | $(1,3,2,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
