Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 26 + \left(32 a + 9\right)\cdot 41 + \left(21 a + 30\right)\cdot 41^{2} + \left(7 a + 11\right)\cdot 41^{3} + \left(18 a + 7\right)\cdot 41^{4} + \left(27 a + 35\right)\cdot 41^{5} + \left(19 a + 22\right)\cdot 41^{6} + \left(9 a + 37\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 17\cdot 41 + 10\cdot 41^{2} + 37\cdot 41^{3} + 36\cdot 41^{4} + 19\cdot 41^{5} + 26\cdot 41^{6} + 23\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 31 + \left(8 a + 37\right)\cdot 41 + \left(19 a + 21\right)\cdot 41^{2} + \left(33 a + 12\right)\cdot 41^{3} + \left(22 a + 13\right)\cdot 41^{4} + \left(13 a + 17\right)\cdot 41^{5} + \left(21 a + 13\right)\cdot 41^{6} + \left(31 a + 5\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 34\cdot 41 + 25\cdot 41^{2} + 36\cdot 41^{3} + 24\cdot 41^{4} + 40\cdot 41^{5} + 36\cdot 41^{6} + 5\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 9 + \left(7 a + 14\right)\cdot 41 + \left(25 a + 32\right)\cdot 41^{2} + \left(33 a + 24\right)\cdot 41^{3} + \left(7 a + 10\right)\cdot 41^{4} + \left(20 a + 10\right)\cdot 41^{5} + 21\cdot 41^{6} + \left(21 a + 12\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 33 + \left(32 a + 8\right)\cdot 41 + \left(12 a + 30\right)\cdot 41^{2} + \left(2 a + 34\right)\cdot 41^{3} + \left(19 a + 7\right)\cdot 41^{4} + \left(a + 37\right)\cdot 41^{5} + \left(36 a + 7\right)\cdot 41^{6} + \left(18 a + 12\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a + 29 + \left(33 a + 3\right)\cdot 41 + \left(15 a + 18\right)\cdot 41^{2} + \left(7 a + 18\right)\cdot 41^{3} + 33 a\cdot 41^{4} + \left(20 a + 22\right)\cdot 41^{5} + \left(40 a + 2\right)\cdot 41^{6} + \left(19 a + 34\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a + 35 + \left(8 a + 37\right)\cdot 41 + \left(28 a + 35\right)\cdot 41^{2} + \left(38 a + 28\right)\cdot 41^{3} + \left(21 a + 21\right)\cdot 41^{4} + \left(39 a + 22\right)\cdot 41^{5} + \left(4 a + 32\right)\cdot 41^{6} + \left(22 a + 32\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,3)(2,5,7)$ |
| $(1,7)(2,4)(3,5)(6,8)$ |
| $(1,4,7,2)(3,6,5,8)$ |
| $(1,3,7,5)(2,6,4,8)$ |
| $(1,2)(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,2)(3,5)(4,7)$ | $0$ |
| $8$ | $3$ | $(1,8,2)(4,7,6)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,5)(2,6,4,8)$ | $0$ |
| $8$ | $6$ | $(1,4,8,7,2,6)(3,5)$ | $1$ |
| $6$ | $8$ | $(1,5,2,6,7,3,4,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,3,2,8,7,5,4,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
