Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{3} + 2 x + 11 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a + 10 + \left(9 a^{2} + 11 a + 8\right)\cdot 13 + \left(11 a^{2} + 10 a\right)\cdot 13^{2} + \left(7 a^{2} + 5 a + 8\right)\cdot 13^{3} + \left(4 a^{2} + 2 a + 6\right)\cdot 13^{4} + \left(12 a + 7\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a^{2} + 7 a + 8 + \left(2 a + 1\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(9 a^{2} + 6 a\right)\cdot 13^{3} + 10\cdot 13^{4} + \left(9 a^{2} + 5 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a^{2} + 6 a + 7 + \left(8 a^{2} + 4 a + 12\right)\cdot 13 + \left(a^{2} + a\right)\cdot 13^{2} + \left(11 a^{2} + 7\right)\cdot 13^{3} + \left(9 a^{2} + 4\right)\cdot 13^{4} + \left(11 a^{2} + 12 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a^{2} + 12 a + 12 + \left(9 a^{2} + 2 a + 5\right)\cdot 13 + \left(4 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(11 a^{2} + 8 a + 11\right)\cdot 13^{3} + \left(4 a^{2} + 11 a + 10\right)\cdot 13^{4} + \left(2 a^{2} + 5 a + 2\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a^{2} + 8 a + 12 + \left(7 a^{2} + 5 a + 11\right)\cdot 13 + \left(6 a^{2} + 9 a + 11\right)\cdot 13^{2} + \left(3 a^{2} + 4 a + 9\right)\cdot 13^{3} + \left(11 a^{2} + a + 10\right)\cdot 13^{4} + \left(11 a^{2} + 8 a + 6\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 5\cdot 13 + 9\cdot 13^{2} + 4\cdot 13^{3} + 11\cdot 13^{4} + 5\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 8 a^{2} + 5 a + 12 + \left(3 a^{2} + 12 a + 5\right)\cdot 13 + \left(a^{2} + 8\right)\cdot 13^{2} + \left(9 a^{2} + a + 9\right)\cdot 13^{3} + \left(7 a^{2} + 10 a + 10\right)\cdot 13^{4} + \left(3 a^{2} + 8 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(3,5)(6,7)$ |
| $(1,4,6,3)(2,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $56$ | $3$ | $(1,6,5)(2,3,4)$ | $0$ |
| $42$ | $4$ | $(1,4,6,3)(2,7)$ | $1$ |
| $24$ | $7$ | $(1,4,7,2,6,5,3)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ | $7$ | $(1,2,3,7,5,4,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
