Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 29 + 16\cdot 41 + 10\cdot 41^{2} + 8\cdot 41^{3} + 21\cdot 41^{4} + 22\cdot 41^{5} + 12\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 22\cdot 41 + 41^{2} + 16\cdot 41^{3} + 38\cdot 41^{4} + 37\cdot 41^{5} + 33\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 24 + \left(35 a + 38\right)\cdot 41 + \left(25 a + 22\right)\cdot 41^{2} + \left(2 a + 32\right)\cdot 41^{3} + \left(16 a + 31\right)\cdot 41^{4} + \left(26 a + 13\right)\cdot 41^{5} + \left(5 a + 21\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 29 + \left(28 a + 30\right)\cdot 41 + \left(11 a + 25\right)\cdot 41^{2} + \left(15 a + 7\right)\cdot 41^{3} + \left(18 a + 19\right)\cdot 41^{4} + \left(12 a + 3\right)\cdot 41^{5} + \left(18 a + 29\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 6 + \left(20 a + 37\right)\cdot 41 + \left(2 a + 29\right)\cdot 41^{2} + \left(21 a + 11\right)\cdot 41^{3} + \left(25 a + 33\right)\cdot 41^{4} + \left(26 a + 27\right)\cdot 41^{5} + \left(40 a + 5\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 30 + \left(5 a + 18\right)\cdot 41 + \left(15 a + 24\right)\cdot 41^{2} + \left(38 a + 14\right)\cdot 41^{3} + \left(24 a + 36\right)\cdot 41^{4} + \left(14 a + 35\right)\cdot 41^{5} + \left(35 a + 11\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 a + 7 + \left(20 a + 2\right)\cdot 41 + \left(38 a + 17\right)\cdot 41^{2} + \left(19 a + 31\right)\cdot 41^{3} + \left(15 a + 6\right)\cdot 41^{4} + 14 a\cdot 41^{5} + 19\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 3 a + 20 + \left(12 a + 38\right)\cdot 41 + \left(29 a + 31\right)\cdot 41^{2} + 25 a\cdot 41^{3} + \left(22 a + 18\right)\cdot 41^{4} + \left(28 a + 22\right)\cdot 41^{5} + \left(22 a + 30\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,7,6)$ |
| $(1,4)(2,3)(5,7)$ |
| $(1,3,7)(2,4,5)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,7,2,5)(3,8,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,4)(2,3)(5,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,8,4)(2,6,3)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,2,5)(3,8,4,6)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,3,8,2,4,6)(5,7)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,5,4,6,2,7,3,8)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,4,8,2,5,3,6)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
