Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 33\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + 36 + \left(17 a + 40\right)\cdot 41 + \left(19 a + 22\right)\cdot 41^{2} + \left(3 a + 3\right)\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + \left(5 a + 19\right)\cdot 41 + \left(29 a + 2\right)\cdot 41^{2} + \left(30 a + 21\right)\cdot 41^{3} + \left(20 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 40 + \left(35 a + 8\right)\cdot 41 + \left(11 a + 2\right)\cdot 41^{2} + \left(10 a + 2\right)\cdot 41^{3} + \left(20 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 30 + \left(9 a + 29\right)\cdot 41 + \left(4 a + 21\right)\cdot 41^{2} + \left(30 a + 22\right)\cdot 41^{3} + \left(3 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 34 + \left(23 a + 38\right)\cdot 41 + \left(21 a + 22\right)\cdot 41^{2} + \left(37 a + 35\right)\cdot 41^{3} + \left(40 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 20 + \left(31 a + 34\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(10 a + 26\right)\cdot 41^{3} + \left(37 a + 1\right)\cdot 41^{4} +O\left(41^{ 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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
