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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 49\cdot 53 + 46\cdot 53^{2} + 25\cdot 53^{3} + 37\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 3 + \left(6 a + 49\right)\cdot 53 + \left(39 a + 28\right)\cdot 53^{2} + \left(13 a + 49\right)\cdot 53^{3} + \left(18 a + 22\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a + 51 + \left(22 a + 35\right)\cdot 53 + \left(7 a + 3\right)\cdot 53^{2} + \left(3 a + 34\right)\cdot 53^{3} + \left(35 a + 20\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 15 + \left(30 a + 30\right)\cdot 53 + \left(45 a + 10\right)\cdot 53^{2} + \left(49 a + 39\right)\cdot 53^{3} + \left(17 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 33\cdot 53 + 48\cdot 53^{2} + 50\cdot 53^{3} + 49\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 27 + \left(46 a + 14\right)\cdot 53 + \left(13 a + 20\right)\cdot 53^{2} + \left(39 a + 12\right)\cdot 53^{3} + \left(34 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(3,4)$ | $0$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $1$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
