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