Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 9\cdot 41 + 34\cdot 41^{2} + 6\cdot 41^{3} + 35\cdot 41^{4} + 11\cdot 41^{5} + 28\cdot 41^{6} + 15\cdot 41^{7} + 22\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 1 + \left(20 a + 25\right)\cdot 41 + \left(24 a + 18\right)\cdot 41^{2} + \left(33 a + 26\right)\cdot 41^{3} + \left(33 a + 38\right)\cdot 41^{4} + \left(33 a + 9\right)\cdot 41^{5} + \left(29 a + 16\right)\cdot 41^{6} + \left(29 a + 29\right)\cdot 41^{7} + \left(14 a + 25\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 11 + \left(20 a + 28\right)\cdot 41 + \left(16 a + 30\right)\cdot 41^{2} + \left(7 a + 20\right)\cdot 41^{3} + \left(7 a + 24\right)\cdot 41^{4} + \left(7 a + 36\right)\cdot 41^{5} + \left(11 a + 30\right)\cdot 41^{6} + \left(11 a + 6\right)\cdot 41^{7} + \left(26 a + 40\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + \left(31 a + 10\right)\cdot 41 + \left(4 a + 1\right)\cdot 41^{2} + \left(34 a + 31\right)\cdot 41^{3} + \left(5 a + 12\right)\cdot 41^{4} + \left(13 a + 29\right)\cdot 41^{5} + \left(14 a + 20\right)\cdot 41^{6} + \left(9 a + 6\right)\cdot 41^{7} + \left(38 a + 23\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 32 + \left(9 a + 26\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(6 a + 5\right)\cdot 41^{3} + \left(35 a + 37\right)\cdot 41^{4} + \left(27 a + 21\right)\cdot 41^{5} + \left(26 a + 9\right)\cdot 41^{6} + \left(31 a + 20\right)\cdot 41^{7} + \left(2 a + 5\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + \left(23 a + 8\right)\cdot 41 + \left(11 a + 36\right)\cdot 41^{2} + \left(17 a + 23\right)\cdot 41^{3} + \left(9 a + 2\right)\cdot 41^{4} + \left(36 a + 12\right)\cdot 41^{5} + \left(13 a + 20\right)\cdot 41^{6} + \left(26 a + 11\right)\cdot 41^{7} + \left(4 a + 31\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 a + 24 + \left(17 a + 28\right)\cdot 41 + \left(29 a + 6\right)\cdot 41^{2} + \left(23 a + 23\right)\cdot 41^{3} + \left(31 a + 13\right)\cdot 41^{4} + \left(4 a + 29\right)\cdot 41^{5} + \left(27 a + 25\right)\cdot 41^{6} + \left(14 a + 35\right)\cdot 41^{7} + \left(36 a + 18\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 7 + 28\cdot 41 + 11\cdot 41^{2} + 26\cdot 41^{3} + 40\cdot 41^{4} + 12\cdot 41^{5} + 12\cdot 41^{6} + 38\cdot 41^{7} + 37\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,4)(3,8,7)$ |
| $(1,5,8,2)(3,7,4,6)$ |
| $(1,3)(4,8)(6,7)$ |
| $(1,8)(2,5)(3,4)(6,7)$ |
| $(1,6,8,7)(2,3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
| $12$ | $2$ | $(1,3)(4,8)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,5,7)(2,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,5,8,2)(3,7,4,6)$ | $0$ |
| $8$ | $6$ | $(1,8)(2,3,7,5,4,6)$ | $1$ |
| $6$ | $8$ | $(1,7,3,5,8,6,4,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,6,3,2,8,7,4,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
