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