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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 24 + \left(44 a + 49\right)\cdot 61 + \left(30 a + 30\right)\cdot 61^{2} + \left(11 a + 15\right)\cdot 61^{3} + \left(15 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a + 34 + \left(44 a + 6\right)\cdot 61 + \left(30 a + 44\right)\cdot 61^{2} + \left(11 a + 17\right)\cdot 61^{3} + \left(15 a + 51\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 22 + \left(44 a + 12\right)\cdot 61 + \left(30 a + 37\right)\cdot 61^{2} + \left(11 a + 56\right)\cdot 61^{3} + \left(15 a + 6\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 2 + \left(16 a + 22\right)\cdot 61 + \left(30 a + 30\right)\cdot 61^{2} + \left(49 a + 59\right)\cdot 61^{3} + \left(45 a + 54\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 53 + \left(16 a + 3\right)\cdot 61 + \left(30 a + 17\right)\cdot 61^{2} + \left(49 a + 57\right)\cdot 61^{3} + 45 a\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 51 + \left(16 a + 27\right)\cdot 61 + \left(30 a + 23\right)\cdot 61^{2} + \left(49 a + 37\right)\cdot 61^{3} + \left(45 a + 10\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,4)(3,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,2,5,3,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,3,5,2,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
