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:
| $r_{ 1 }$ |
$=$ |
$ 32 + 52\cdot 53 + 38\cdot 53^{2} + 19\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 5 + \left(5 a + 34\right)\cdot 53 + \left(3 a + 52\right)\cdot 53^{2} + \left(49 a + 16\right)\cdot 53^{3} + \left(4 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 37\cdot 53 + 36\cdot 53^{2} + 24\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a + 6 + \left(47 a + 17\right)\cdot 53 + \left(49 a + 6\right)\cdot 53^{2} + \left(3 a + 51\right)\cdot 53^{3} + \left(48 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 35 + \left(34 a + 30\right)\cdot 53 + \left(28 a + 51\right)\cdot 53^{2} + \left(36 a + 43\right)\cdot 53^{3} + \left(50 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 17 + \left(18 a + 40\right)\cdot 53 + \left(24 a + 25\right)\cdot 53^{2} + \left(16 a + 2\right)\cdot 53^{3} + \left(2 a + 5\right)\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
