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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + \left(24 a + 53\right)\cdot 61 + \left(60 a + 5\right)\cdot 61^{2} + \left(36 a + 48\right)\cdot 61^{3} + \left(32 a + 31\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 16 + 36 a\cdot 61 + 42\cdot 61^{2} + \left(24 a + 24\right)\cdot 61^{3} + \left(28 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 10 + \left(24 a + 10\right)\cdot 61 + \left(60 a + 19\right)\cdot 61^{2} + \left(36 a + 50\right)\cdot 61^{3} + \left(32 a + 24\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 26 + \left(36 a + 18\right)\cdot 61 + 55\cdot 61^{2} + \left(24 a + 26\right)\cdot 61^{3} + \left(28 a + 20\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 59 + \left(24 a + 15\right)\cdot 61 + \left(60 a + 12\right)\cdot 61^{2} + \left(36 a + 28\right)\cdot 61^{3} + \left(32 a + 41\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 a + 14 + \left(36 a + 24\right)\cdot 61 + 48\cdot 61^{2} + \left(24 a + 4\right)\cdot 61^{3} + \left(28 a + 37\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,5,3)(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,3,2,5,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,5,2,3,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
