Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 12\cdot 13 + \left(7 a + 3\right)\cdot 13^{2} + \left(a + 8\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 2 + \left(2 a + 2\right)\cdot 13 + \left(11 a + 11\right)\cdot 13^{2} + \left(7 a + 7\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 12 + 12 a\cdot 13 + \left(5 a + 10\right)\cdot 13^{2} + \left(11 a + 2\right)\cdot 13^{3} + \left(8 a + 7\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 1 + \left(10 a + 6\right)\cdot 13 + \left(a + 6\right)\cdot 13^{2} + \left(5 a + 4\right)\cdot 13^{3} + \left(8 a + 1\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 9\cdot 13 + 2\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 2 a\cdot 13 + \left(2 a + 10\right)\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + 2 a\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 a + 7 + \left(10 a + 8\right)\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(2 a + 11\right)\cdot 13^{3} + \left(10 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,7)(2,5)(4,6)$ |
| $(1,5)(2,6)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $2$ | $7$ | $(1,2,4,3,6,5,7)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,4,6,7,2,3,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,3,7,4,5,2,6)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
