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:
| $r_{ 1 }$ |
$=$ |
$ 39 a^{2} + 32 a + 23 + \left(2 a^{2} + a + 28\right)\cdot 41 + \left(4 a^{2} + 12 a + 12\right)\cdot 41^{2} + \left(7 a^{2} + 18\right)\cdot 41^{3} + \left(30 a^{2} + 37 a + 13\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a^{2} + 29 a + 37 + \left(35 a^{2} + 12 a + 32\right)\cdot 41 + \left(3 a^{2} + 4 a + 37\right)\cdot 41^{2} + \left(22 a^{2} + 12 a + 34\right)\cdot 41^{3} + \left(27 a^{2} + 16 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a^{2} + 31 a + 10 + \left(34 a^{2} + 4 a + 36\right)\cdot 41 + \left(4 a^{2} + 5 a + 26\right)\cdot 41^{2} + \left(10 a^{2} + 21 a + 6\right)\cdot 41^{3} + \left(11 a^{2} + 17 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 22 a + 22 + \left(16 a^{2} + 33 a + 29\right)\cdot 41 + \left(a^{2} + 10 a + 37\right)\cdot 41^{2} + \left(4 a^{2} + a + 9\right)\cdot 41^{3} + \left(32 a^{2} + 18 a + 21\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a^{2} + 38 a + 16 + \left(40 a^{2} + 27 a + 22\right)\cdot 41 + \left(19 a^{2} + 2 a + 7\right)\cdot 41^{2} + \left(38 a^{2} + a + 32\right)\cdot 41^{3} + \left(26 a^{2} + 33 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a^{2} + 15 a + 22 + \left(6 a^{2} + 13\right)\cdot 41 + \left(17 a^{2} + 34 a + 19\right)\cdot 41^{2} + \left(21 a^{2} + 27 a + 34\right)\cdot 41^{3} + \left(27 a^{2} + 32 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a^{2} + 19 a + 40 + \left(3 a^{2} + 34 a + 14\right)\cdot 41 + \left(32 a^{2} + 23 a + 31\right)\cdot 41^{2} + \left(23 a^{2} + 19 a + 15\right)\cdot 41^{3} + \left(40 a^{2} + 27 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 a^{2} + 10 a + 14 + \left(12 a^{2} + 13 a + 27\right)\cdot 41 + \left(33 a^{2} + 16 a + 31\right)\cdot 41^{2} + \left(17 a^{2} + 25 a + 32\right)\cdot 41^{3} + \left(22 a^{2} + 10 a + 14\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 4 a^{2} + 9 a + 22 + \left(12 a^{2} + 35 a + 40\right)\cdot 41 + \left(6 a^{2} + 13 a + 40\right)\cdot 41^{2} + \left(19 a^{2} + 14 a + 19\right)\cdot 41^{3} + \left(27 a^{2} + 12 a + 4\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(2,4)(5,8)(6,9)$ |
| $(1,7,3)(2,5,6)(4,9,8)$ |
| $(1,8,2,7,4,5,3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,3)(2,4)(5,8)(6,9)$ | $0$ |
| $2$ | $3$ | $(1,7,3)(2,5,6)(4,9,8)$ | $-1$ |
| $2$ | $9$ | $(1,8,2,7,4,5,3,9,6)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,2,4,3,6,8,7,5,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,4,6,7,9,2,3,8,5)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
