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:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 48 + \left(9 a + 39\right)\cdot 53 + \left(17 a + 18\right)\cdot 53^{2} + \left(36 a + 40\right)\cdot 53^{3} + \left(35 a + 25\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 40 + \left(43 a + 22\right)\cdot 53 + \left(35 a + 36\right)\cdot 53^{2} + \left(16 a + 43\right)\cdot 53^{3} + \left(17 a + 6\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 27 + \left(9 a + 26\right)\cdot 53 + \left(17 a + 30\right)\cdot 53^{2} + \left(36 a + 21\right)\cdot 53^{3} + \left(35 a + 6\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a + 12 + \left(9 a + 45\right)\cdot 53 + \left(17 a + 47\right)\cdot 53^{2} + \left(36 a + 10\right)\cdot 53^{3} + \left(35 a + 20\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 8 + \left(43 a + 36\right)\cdot 53 + \left(35 a + 24\right)\cdot 53^{2} + \left(16 a + 9\right)\cdot 53^{3} + \left(17 a + 26\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 25 + \left(43 a + 41\right)\cdot 53 + 35 a\cdot 53^{2} + \left(16 a + 33\right)\cdot 53^{3} + \left(17 a + 20\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,3,5,4,2)$ |
| $(1,5)(2,3)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,3,5,4,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,4,5,3,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
