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:
| $r_{ 1 }$ |
$=$ |
$ 8 + 42\cdot 53 + 47\cdot 53^{2} + 14\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 43 + \left(44 a + 9\right)\cdot 53 + \left(37 a + 9\right)\cdot 53^{2} + \left(8 a + 16\right)\cdot 53^{3} + \left(30 a + 39\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 45\cdot 53 + 12\cdot 53^{2} + 19\cdot 53^{3} + 10\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 a + 4 + \left(36 a + 10\right)\cdot 53 + \left(40 a + 3\right)\cdot 53^{2} + \left(26 a + 41\right)\cdot 53^{3} + \left(13 a + 43\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 45 + \left(16 a + 1\right)\cdot 53 + \left(12 a + 23\right)\cdot 53^{2} + \left(26 a + 1\right)\cdot 53^{3} + \left(39 a + 18\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 16 + \left(8 a + 49\right)\cdot 53 + \left(15 a + 9\right)\cdot 53^{2} + \left(44 a + 13\right)\cdot 53^{3} + \left(22 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,4,6,5)$ |
| $(1,3)(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $-2$ |
| $20$ | $3$ | $(1,3,6)(2,4,5)$ | $0$ |
| $30$ | $4$ | $(1,6,5,4)$ | $0$ |
| $24$ | $5$ | $(1,4,3,2,6)$ | $1$ |
| $20$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
