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:
| $r_{ 1 }$ |
$=$ |
$ 41 a + 40 + \left(37 a + 42\right)\cdot 53 + \left(48 a + 11\right)\cdot 53^{2} + \left(25 a + 37\right)\cdot 53^{3} + 21\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 45 + \left(15 a + 46\right)\cdot 53 + \left(4 a + 9\right)\cdot 53^{2} + \left(27 a + 39\right)\cdot 53^{3} + \left(52 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 51 + \left(21 a + 12\right)\cdot 53 + \left(27 a + 37\right)\cdot 53^{2} + \left(28 a + 18\right)\cdot 53^{3} + \left(34 a + 42\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 3\cdot 53 + 47\cdot 53^{2} + 38\cdot 53^{3} + 34\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 42 + \left(31 a + 33\right)\cdot 53 + \left(25 a + 19\right)\cdot 53^{2} + \left(24 a + 52\right)\cdot 53^{3} + \left(18 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 18\cdot 53 + 33\cdot 53^{2} + 25\cdot 53^{3} + 16\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,2)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,5,2,3)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $0$ |
| $12$ | $6$ | $(1,2)(3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
