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:
| $r_{ 1 }$ |
$=$ |
$ 48 a + 16 + \left(4 a + 4\right)\cdot 53 + \left(46 a + 6\right)\cdot 53^{2} + \left(38 a + 39\right)\cdot 53^{3} + \left(24 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a + 14 + \left(51 a + 30\right)\cdot 53 + \left(17 a + 47\right)\cdot 53^{2} + \left(31 a + 31\right)\cdot 53^{3} + \left(4 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 13 + \left(41 a + 24\right)\cdot 53 + \left(38 a + 5\right)\cdot 53^{2} + \left(50 a + 33\right)\cdot 53^{3} + \left(14 a + 8\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 49 + \left(48 a + 28\right)\cdot 53 + \left(6 a + 26\right)\cdot 53^{2} + \left(14 a + 42\right)\cdot 53^{3} + \left(28 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 39 + \left(11 a + 51\right)\cdot 53 + \left(14 a + 12\right)\cdot 53^{2} + \left(2 a + 38\right)\cdot 53^{3} + \left(38 a + 17\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 + 35\cdot 53 + 45\cdot 53^{2} + 46\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 a + 19 + \left(a + 37\right)\cdot 53 + \left(35 a + 14\right)\cdot 53^{2} + \left(21 a + 33\right)\cdot 53^{3} + \left(48 a + 47\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
