Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 8 + \left(30 a + 19\right)\cdot 53 + \left(24 a + 47\right)\cdot 53^{2} + \left(21 a + 12\right)\cdot 53^{3} + \left(39 a + 17\right)\cdot 53^{4} + 7\cdot 53^{5} + \left(13 a + 46\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 a + 4 + \left(30 a + 13\right)\cdot 53 + \left(24 a + 44\right)\cdot 53^{2} + \left(21 a + 31\right)\cdot 53^{3} + \left(39 a + 5\right)\cdot 53^{4} + 29\cdot 53^{5} + \left(13 a + 8\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 33\cdot 53 + 46\cdot 53^{2} + 34\cdot 53^{3} + 27\cdot 53^{4} + 50\cdot 53^{5} + 4\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 45 + \left(22 a + 33\right)\cdot 53 + \left(28 a + 5\right)\cdot 53^{2} + \left(31 a + 40\right)\cdot 53^{3} + \left(13 a + 35\right)\cdot 53^{4} + \left(52 a + 45\right)\cdot 53^{5} + \left(39 a + 6\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 49 + \left(22 a + 39\right)\cdot 53 + \left(28 a + 8\right)\cdot 53^{2} + \left(31 a + 21\right)\cdot 53^{3} + \left(13 a + 47\right)\cdot 53^{4} + \left(52 a + 23\right)\cdot 53^{5} + \left(39 a + 44\right)\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 19\cdot 53 + 6\cdot 53^{2} + 18\cdot 53^{3} + 25\cdot 53^{4} + 2\cdot 53^{5} + 48\cdot 53^{6} +O\left(53^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $1$ |
| $6$ | $4$ | $(1,6,4,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
