Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 19\cdot 61 + 21\cdot 61^{2} + 56\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 60\cdot 61 + 2\cdot 61^{2} + 28\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 47 + \left(27 a + 55\right)\cdot 61 + \left(34 a + 32\right)\cdot 61^{2} + \left(51 a + 56\right)\cdot 61^{3} + \left(56 a + 7\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 51 + \left(47 a + 30\right)\cdot 61 + \left(22 a + 24\right)\cdot 61^{2} + \left(55 a + 59\right)\cdot 61^{3} + \left(27 a + 20\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 56 + \left(13 a + 11\right)\cdot 61 + 38 a\cdot 61^{2} + \left(5 a + 31\right)\cdot 61^{3} + \left(33 a + 54\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a + 3 + \left(33 a + 5\right)\cdot 61 + \left(26 a + 40\right)\cdot 61^{2} + \left(9 a + 12\right)\cdot 61^{3} + \left(4 a + 13\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,5)(2,3,4)$ |
| $(1,4)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(2,6)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,6,5)(2,3,4)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,5,6,4,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,6,2,5,4)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
