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:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 38 + \left(42 a + 34\right)\cdot 61 + \left(47 a + 27\right)\cdot 61^{2} + \left(2 a + 3\right)\cdot 61^{3} + 9 a\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a + 11 + \left(4 a + 50\right)\cdot 61 + \left(8 a + 60\right)\cdot 61^{2} + \left(15 a + 43\right)\cdot 61^{3} + \left(41 a + 2\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 60 + \left(56 a + 5\right)\cdot 61 + \left(52 a + 3\right)\cdot 61^{2} + \left(45 a + 51\right)\cdot 61^{3} + \left(19 a + 28\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 7\cdot 61 + 59\cdot 61^{2} + 3\cdot 61^{3} + 23\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 31 + \left(18 a + 23\right)\cdot 61 + \left(13 a + 32\right)\cdot 61^{2} + \left(58 a + 19\right)\cdot 61^{3} + \left(51 a + 6\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
