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^{3} + x + 35 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 a^{2} + 32 a + 8 + \left(29 a^{2} + 8 a + 28\right)\cdot 41 + \left(4 a^{2} + 3 a + 4\right)\cdot 41^{2} + \left(36 a^{2} + 22 a + 1\right)\cdot 41^{3} + \left(4 a^{2} + 36 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a^{2} + 11 a + 31 + \left(10 a^{2} + 3 a + 15\right)\cdot 41 + \left(7 a^{2} + 40 a + 6\right)\cdot 41^{2} + \left(9 a^{2} + 36 a + 24\right)\cdot 41^{3} + \left(23 a^{2} + 11 a + 23\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a^{2} + 37 a + 4 + \left(33 a^{2} + 10 a + 9\right)\cdot 41 + \left(19 a^{2} + 37 a + 31\right)\cdot 41^{2} + \left(20 a^{2} + 14 a + 34\right)\cdot 41^{3} + \left(14 a^{2} + 2 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a^{2} + 5 a + 33 + \left(14 a^{2} + 20 a\right)\cdot 41 + \left(18 a^{2} + 28 a + 20\right)\cdot 41^{2} + \left(28 a^{2} + 15 a + 34\right)\cdot 41^{3} + \left(4 a^{2} + 37 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a^{2} + 36 a + 8 + \left(10 a^{2} + 14 a + 25\right)\cdot 41 + \left(10 a^{2} + 4 a + 14\right)\cdot 41^{2} + \left(22 a^{2} + a + 30\right)\cdot 41^{3} + \left(23 a^{2} + 13 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a^{2} + 35 + \left(15 a^{2} + 6 a + 28\right)\cdot 41 + \left(12 a^{2} + 8 a + 29\right)\cdot 41^{2} + \left(31 a^{2} + 24 a + 22\right)\cdot 41^{3} + \left(12 a^{2} + 31 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 13 a + 4 + \left(23 a^{2} + 31 a + 16\right)\cdot 41 + \left(21 a^{2} + 27 a + 32\right)\cdot 41^{2} + \left(39 a^{2} + 8 a + 33\right)\cdot 41^{3} + \left(27 a^{2} + 36 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ a^{2} + 39 a + \left(a^{2} + 28 a + 23\right)\cdot 41 + \left(29 a^{2} + 38 a + 34\right)\cdot 41^{2} + \left(36 a^{2} + 22 a + 28\right)\cdot 41^{3} + \left(12 a^{2} + 33 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 40 a^{2} + 32 a + 3 + \left(24 a^{2} + 39 a + 17\right)\cdot 41 + \left(40 a^{2} + 16 a + 31\right)\cdot 41^{2} + \left(21 a^{2} + 17 a + 35\right)\cdot 41^{3} + \left(39 a^{2} + 2 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(2,7)(5,6)(8,9)$ |
| $(1,2,8)(3,9,7)(4,6,5)$ |
| $(1,7,4,2,3,6,8,9,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,7)(5,6)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,2,8)(3,9,7)(4,6,5)$ | $-1$ |
| $2$ | $9$ | $(1,7,4,2,3,6,8,9,5)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,4,3,8,5,7,2,6,9)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,3,5,2,9,4,8,7,6)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
