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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 13 + \left(17 a + 29\right)\cdot 31 + \left(18 a + 1\right)\cdot 31^{2} + \left(8 a + 7\right)\cdot 31^{3} + \left(23 a + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 21 + \left(10 a + 3\right)\cdot 31 + \left(14 a + 18\right)\cdot 31^{2} + \left(16 a + 3\right)\cdot 31^{3} + \left(6 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 1 + \left(13 a + 21\right)\cdot 31 + 25\cdot 31^{2} + \left(19 a + 23\right)\cdot 31^{3} + \left(24 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 8 + \left(13 a + 20\right)\cdot 31 + \left(12 a + 21\right)\cdot 31^{2} + \left(22 a + 5\right)\cdot 31^{3} + \left(7 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 6 + \left(17 a + 30\right)\cdot 31 + \left(30 a + 12\right)\cdot 31^{2} + \left(11 a + 30\right)\cdot 31^{3} + \left(6 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 9 + 20 a\cdot 31 + \left(16 a + 5\right)\cdot 31^{2} + \left(14 a + 22\right)\cdot 31^{3} + \left(24 a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 + 19\cdot 31 + 7\cdot 31^{2} + 9\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$-5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
