Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 54 + \left(31 a + 59\right)\cdot 61 + \left(45 a + 13\right)\cdot 61^{2} + \left(26 a + 31\right)\cdot 61^{3} + \left(12 a + 22\right)\cdot 61^{4} + \left(58 a + 40\right)\cdot 61^{5} + \left(16 a + 21\right)\cdot 61^{6} + \left(16 a + 51\right)\cdot 61^{7} + \left(45 a + 31\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 5\cdot 61 + 29\cdot 61^{2} + 44\cdot 61^{3} + 16\cdot 61^{4} + 31\cdot 61^{5} + 6\cdot 61^{6} + 52\cdot 61^{7} + 50\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 8 + \left(38 a + 22\right)\cdot 61 + \left(9 a + 6\right)\cdot 61^{2} + \left(54 a + 53\right)\cdot 61^{3} + \left(2 a + 44\right)\cdot 61^{4} + \left(7 a + 20\right)\cdot 61^{5} + 7\cdot 61^{6} + \left(11 a + 54\right)\cdot 61^{7} + \left(58 a + 29\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 37\cdot 61 + 5\cdot 61^{2} + 5\cdot 61^{3} + 36\cdot 61^{4} + 40\cdot 61^{5} + 44\cdot 61^{6} + 31\cdot 61^{7} + 54\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a + 57 + \left(29 a + 26\right)\cdot 61 + \left(15 a + 28\right)\cdot 61^{2} + \left(34 a + 12\right)\cdot 61^{3} + \left(48 a + 8\right)\cdot 61^{4} + \left(2 a + 25\right)\cdot 61^{5} + \left(44 a + 41\right)\cdot 61^{6} + \left(44 a + 50\right)\cdot 61^{7} + \left(15 a + 60\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 38 + \left(22 a + 30\right)\cdot 61 + \left(51 a + 38\right)\cdot 61^{2} + \left(6 a + 36\right)\cdot 61^{3} + \left(58 a + 54\right)\cdot 61^{4} + \left(53 a + 24\right)\cdot 61^{5} + 60 a\cdot 61^{6} + \left(49 a + 4\right)\cdot 61^{7} + \left(2 a + 16\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2,5)(3,4,6)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,4)$ |
$-1$ |
| $6$ |
$2$ |
$(2,5)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,3,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,3,5)(2,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,6,3,2,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
