Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 22 + \left(24 a + 21\right)\cdot 31 + \left(17 a + 12\right)\cdot 31^{2} + \left(7 a + 15\right)\cdot 31^{3} + \left(8 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 19 + \left(6 a + 29\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(29 a + 11\right)\cdot 31^{3} + \left(18 a + 29\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 29 + \left(11 a + 6\right)\cdot 31 + \left(3 a + 29\right)\cdot 31^{2} + \left(17 a + 23\right)\cdot 31^{3} + \left(12 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 30 + \left(19 a + 13\right)\cdot 31 + \left(27 a + 24\right)\cdot 31^{2} + \left(13 a + 23\right)\cdot 31^{3} + \left(18 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 6 + \left(24 a + 2\right)\cdot 31 + \left(18 a + 23\right)\cdot 31^{2} + \left(a + 26\right)\cdot 31^{3} + \left(12 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 30 + \left(6 a + 3\right)\cdot 31 + \left(13 a + 24\right)\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + \left(22 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 + 14\cdot 31 + 5\cdot 31^{2} + 10\cdot 31^{3} + 14\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
