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:
| $r_{ 1 }$ |
$=$ |
$ 31 + 53 + 45\cdot 53^{2} + 49\cdot 53^{3} + 17\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 a + 40 + \left(14 a + 14\right)\cdot 53 + 40 a\cdot 53^{2} + \left(40 a + 38\right)\cdot 53^{3} + \left(7 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 23 + \left(49 a + 12\right)\cdot 53 + \left(22 a + 1\right)\cdot 53^{2} + \left(20 a + 9\right)\cdot 53^{3} + \left(32 a + 18\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 40\cdot 53 + 27\cdot 53^{2} + 45\cdot 53^{3} + 24\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 10 + 38 a\cdot 53 + \left(12 a + 41\right)\cdot 53^{2} + \left(12 a + 1\right)\cdot 53^{3} + \left(45 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 26 + \left(3 a + 36\right)\cdot 53 + \left(30 a + 43\right)\cdot 53^{2} + \left(32 a + 14\right)\cdot 53^{3} + \left(20 a + 21\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.
