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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 40 + \left(25 a + 33\right)\cdot 41 + \left(16 a + 17\right)\cdot 41^{2} + \left(22 a + 35\right)\cdot 41^{3} + \left(9 a + 1\right)\cdot 41^{4} + \left(3 a + 13\right)\cdot 41^{5} + \left(a + 19\right)\cdot 41^{6} + \left(31 a + 23\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 35 + 25 a\cdot 41 + \left(16 a + 40\right)\cdot 41^{2} + \left(22 a + 36\right)\cdot 41^{3} + \left(9 a + 32\right)\cdot 41^{4} + \left(3 a + 27\right)\cdot 41^{5} + \left(a + 21\right)\cdot 41^{6} + \left(31 a + 7\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 6\cdot 41^{2} + 18\cdot 41^{3} + 15\cdot 41^{4} + 26\cdot 41^{5} + 23\cdot 41^{6} + 11\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 26 + \left(11 a + 35\right)\cdot 41 + \left(a + 3\right)\cdot 41^{2} + \left(37 a + 27\right)\cdot 41^{3} + \left(12 a + 19\right)\cdot 41^{4} + \left(5 a + 39\right)\cdot 41^{5} + \left(18 a + 36\right)\cdot 41^{6} + \left(36 a + 15\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 7 + \left(15 a + 40\right)\cdot 41 + 24 a\cdot 41^{2} + \left(18 a + 4\right)\cdot 41^{3} + \left(31 a + 8\right)\cdot 41^{4} + \left(37 a + 13\right)\cdot 41^{5} + \left(39 a + 19\right)\cdot 41^{6} + \left(9 a + 33\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 2 + \left(15 a + 7\right)\cdot 41 + \left(24 a + 23\right)\cdot 41^{2} + \left(18 a + 5\right)\cdot 41^{3} + \left(31 a + 39\right)\cdot 41^{4} + \left(37 a + 27\right)\cdot 41^{5} + \left(39 a + 21\right)\cdot 41^{6} + \left(9 a + 17\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 + 40\cdot 41 + 34\cdot 41^{2} + 22\cdot 41^{3} + 25\cdot 41^{4} + 14\cdot 41^{5} + 17\cdot 41^{6} + 29\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a + 16 + \left(29 a + 5\right)\cdot 41 + \left(39 a + 37\right)\cdot 41^{2} + \left(3 a + 13\right)\cdot 41^{3} + \left(28 a + 21\right)\cdot 41^{4} + \left(35 a + 1\right)\cdot 41^{5} + \left(22 a + 4\right)\cdot 41^{6} + \left(4 a + 25\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,6,7)(2,4,5,8)$ |
| $(2,3,4)(5,7,8)$ |
| $(2,5)(3,8)(4,7)$ |
| $(1,6)(2,5)(3,7)(4,8)$ |
| $(1,4,6,8)(2,7,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(2,5)(3,8)(4,7)$ | $0$ |
| $8$ | $3$ | $(2,3,4)(5,7,8)$ | $-1$ |
| $6$ | $4$ | $(1,3,6,7)(2,4,5,8)$ | $0$ |
| $8$ | $6$ | $(1,6)(2,7,4,5,3,8)$ | $1$ |
| $6$ | $8$ | $(1,3,2,8,6,7,5,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,7,2,4,6,3,5,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
