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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 37 + \left(45 a + 29\right)\cdot 53 + \left(40 a + 28\right)\cdot 53^{2} + \left(49 a + 50\right)\cdot 53^{3} + \left(52 a + 34\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 35 + \left(29 a + 43\right)\cdot 53 + \left(37 a + 11\right)\cdot 53^{2} + \left(46 a + 39\right)\cdot 53^{3} + \left(35 a + 35\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 3 + \left(7 a + 34\right)\cdot 53 + \left(12 a + 40\right)\cdot 53^{2} + \left(3 a + 49\right)\cdot 53^{3} + 37\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 + 15\cdot 53 + 51\cdot 53^{2} + 52\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 5 + \left(23 a + 36\right)\cdot 53 + \left(15 a + 26\right)\cdot 53^{2} + \left(6 a + 29\right)\cdot 53^{3} + \left(17 a + 26\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 43\cdot 53^{3} + 25\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
