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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 19\cdot 31 + 20\cdot 31^{2} + 2\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 29 + \left(23 a + 28\right)\cdot 31 + 11 a\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 5\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(13 a + 15\right)\cdot 31^{3} + \left(29 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 6 + \left(16 a + 15\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + \left(17 a + 25\right)\cdot 31^{3} + \left(a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 7 + \left(7 a + 25\right)\cdot 31 + 19 a\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(30 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
