Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 27 + \left(38 a + 38\right)\cdot 61 + \left(60 a + 22\right)\cdot 61^{2} + \left(7 a + 10\right)\cdot 61^{3} + \left(47 a + 7\right)\cdot 61^{4} + \left(26 a + 36\right)\cdot 61^{5} + \left(39 a + 15\right)\cdot 61^{6} + \left(54 a + 26\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 16\cdot 61 + 37\cdot 61^{2} + 12\cdot 61^{3} + 14\cdot 61^{4} + 17\cdot 61^{5} + 55\cdot 61^{6} + 25\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 a + 38 + \left(5 a + 57\right)\cdot 61 + \left(15 a + 15\right)\cdot 61^{2} + \left(9 a + 55\right)\cdot 61^{3} + \left(57 a + 42\right)\cdot 61^{4} + \left(35 a + 51\right)\cdot 61^{5} + \left(47 a + 46\right)\cdot 61^{6} + \left(4 a + 36\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 35\cdot 61 + 37\cdot 61^{2} + 36\cdot 61^{3} + 42\cdot 61^{4} + 31\cdot 61^{5} + 39\cdot 61^{6} + 58\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 33 + \left(55 a + 7\right)\cdot 61 + \left(45 a + 25\right)\cdot 61^{2} + \left(51 a + 49\right)\cdot 61^{3} + \left(3 a + 29\right)\cdot 61^{4} + \left(25 a + 30\right)\cdot 61^{5} + \left(13 a + 58\right)\cdot 61^{6} + \left(56 a + 54\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 16 + \left(22 a + 27\right)\cdot 61 + 44\cdot 61^{2} + \left(53 a + 18\right)\cdot 61^{3} + \left(13 a + 46\right)\cdot 61^{4} + \left(34 a + 15\right)\cdot 61^{5} + \left(21 a + 28\right)\cdot 61^{6} + \left(6 a + 41\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,5)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,5,6)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5)(2,3,4,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,2,3,5,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
