Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 2\cdot 29 + 2\cdot 29^{2} + 23\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 2 + \left(22 a + 16\right)\cdot 29 + \left(a + 18\right)\cdot 29^{2} + \left(23 a + 15\right)\cdot 29^{3} + \left(23 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 22 + \left(6 a + 6\right)\cdot 29 + \left(27 a + 5\right)\cdot 29^{2} + \left(5 a + 13\right)\cdot 29^{3} + \left(5 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 17 + \left(20 a + 3\right)\cdot 29 + \left(13 a + 23\right)\cdot 29^{2} + \left(28 a + 5\right)\cdot 29^{3} + \left(27 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 12 + \left(12 a + 27\right)\cdot 29 + \left(9 a + 9\right)\cdot 29^{2} + \left(16 a + 27\right)\cdot 29^{3} + \left(8 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 12 + \left(8 a + 22\right)\cdot 29 + \left(15 a + 12\right)\cdot 29^{2} + 18\cdot 29^{3} + \left(a + 3\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 a + 21 + \left(16 a + 8\right)\cdot 29 + \left(19 a + 15\right)\cdot 29^{2} + \left(12 a + 12\right)\cdot 29^{3} + \left(20 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,6)(2,4)(3,7)$ |
| $(1,3)(2,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $7$ |
$2$ |
$(1,6)(2,4)(3,7)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$7$ |
$(1,5,6,3,2,4,7)$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ |
$7$ |
$(1,6,2,7,5,3,4)$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ |
$7$ |
$(1,3,7,6,4,5,2)$ |
$\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
