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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 10 + \left(24 a + 21\right)\cdot 53 + \left(24 a + 20\right)\cdot 53^{2} + \left(8 a + 51\right)\cdot 53^{3} + \left(41 a + 30\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 a + 39 + \left(29 a + 19\right)\cdot 53 + \left(12 a + 24\right)\cdot 53^{2} + \left(21 a + 8\right)\cdot 53^{3} + \left(39 a + 36\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 25 + \left(23 a + 8\right)\cdot 53 + \left(40 a + 45\right)\cdot 53^{2} + \left(31 a + 27\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 50\cdot 53 + 27\cdot 53^{2} + 10\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 a + 34 + \left(28 a + 5\right)\cdot 53 + \left(28 a + 41\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(11 a + 28\right)\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
