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 a + 6 + \left(3 a + 44\right)\cdot 53 + \left(39 a + 5\right)\cdot 53^{2} + \left(28 a + 40\right)\cdot 53^{3} + \left(45 a + 37\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 33 + \left(42 a + 16\right)\cdot 53 + \left(25 a + 12\right)\cdot 53^{2} + \left(4 a + 12\right)\cdot 53^{3} + \left(40 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 28 + \left(49 a + 26\right)\cdot 53 + \left(13 a + 52\right)\cdot 53^{2} + \left(24 a + 9\right)\cdot 53^{3} + \left(7 a + 32\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 48 + \left(10 a + 9\right)\cdot 53 + \left(27 a + 20\right)\cdot 53^{2} + \left(48 a + 4\right)\cdot 53^{3} + 12 a\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 35\cdot 53 + 47\cdot 53^{2} + 2\cdot 53^{3} + 36\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 26\cdot 53 + 20\cdot 53^{2} + 36\cdot 53^{3} + 49\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,4,5,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
