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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 38\cdot 53 + 17\cdot 53^{2} + 9\cdot 53^{3} + 45\cdot 53^{4} + 33\cdot 53^{5} +O\left(53^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 2 + \left(41 a + 51\right)\cdot 53 + \left(44 a + 30\right)\cdot 53^{2} + \left(27 a + 10\right)\cdot 53^{3} + \left(50 a + 51\right)\cdot 53^{4} + \left(32 a + 48\right)\cdot 53^{5} + \left(45 a + 22\right)\cdot 53^{6} + 29 a\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 48 a + 11 + \left(12 a + 43\right)\cdot 53 + \left(10 a + 9\right)\cdot 53^{2} + \left(20 a + 22\right)\cdot 53^{3} + \left(5 a + 50\right)\cdot 53^{4} + \left(11 a + 32\right)\cdot 53^{5} + \left(a + 37\right)\cdot 53^{6} + \left(19 a + 4\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 49 + \left(11 a + 32\right)\cdot 53 + \left(8 a + 9\right)\cdot 53^{2} + \left(25 a + 24\right)\cdot 53^{3} + \left(2 a + 13\right)\cdot 53^{4} + \left(20 a + 24\right)\cdot 53^{5} + \left(7 a + 13\right)\cdot 53^{6} + \left(23 a + 21\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 44 + \left(40 a + 46\right)\cdot 53 + \left(42 a + 37\right)\cdot 53^{2} + \left(32 a + 39\right)\cdot 53^{3} + \left(47 a + 51\right)\cdot 53^{4} + \left(41 a + 18\right)\cdot 53^{5} + \left(51 a + 31\right)\cdot 53^{6} + \left(33 a + 26\right)\cdot 53^{7} +O\left(53^{ 8 }\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.
