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:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 12 + \left(21 a + 30\right)\cdot 53 + \left(34 a + 17\right)\cdot 53^{2} + \left(45 a + 16\right)\cdot 53^{3} + \left(41 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a + 2 + \left(31 a + 11\right)\cdot 53 + \left(18 a + 39\right)\cdot 53^{2} + \left(7 a + 34\right)\cdot 53^{3} + \left(11 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 39 + \left(21 a + 36\right)\cdot 53 + \left(34 a + 28\right)\cdot 53^{2} + \left(45 a + 45\right)\cdot 53^{3} + \left(41 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 28 + \left(31 a + 4\right)\cdot 53 + \left(18 a + 28\right)\cdot 53^{2} + \left(7 a + 5\right)\cdot 53^{3} + \left(11 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 5 + \left(21 a + 51\right)\cdot 53 + \left(34 a + 43\right)\cdot 53^{2} + \left(45 a + 33\right)\cdot 53^{3} + \left(41 a + 44\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 21 + \left(31 a + 25\right)\cdot 53 + \left(18 a + 1\right)\cdot 53^{2} + \left(7 a + 23\right)\cdot 53^{3} + \left(11 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,2,5,4,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,5,4,3,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,3,4,5,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
