Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 a + 37 + \left(18 a + 58\right)\cdot 61 + \left(45 a + 10\right)\cdot 61^{2} + \left(41 a + 53\right)\cdot 61^{3} + \left(50 a + 1\right)\cdot 61^{4} + \left(27 a + 49\right)\cdot 61^{5} + \left(18 a + 39\right)\cdot 61^{6} + \left(20 a + 3\right)\cdot 61^{7} + \left(28 a + 54\right)\cdot 61^{8} + \left(18 a + 31\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 24 + \left(40 a + 42\right)\cdot 61 + \left(21 a + 12\right)\cdot 61^{2} + \left(26 a + 41\right)\cdot 61^{3} + \left(48 a + 51\right)\cdot 61^{4} + \left(36 a + 41\right)\cdot 61^{5} + \left(a + 14\right)\cdot 61^{6} + \left(7 a + 48\right)\cdot 61^{7} + \left(17 a + 5\right)\cdot 61^{8} + \left(56 a + 40\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 + 42\cdot 61 + 44\cdot 61^{2} + 33\cdot 61^{3} + 48\cdot 61^{4} + 45\cdot 61^{5} + 36\cdot 61^{6} + 14\cdot 61^{7} + 52\cdot 61^{8} + 60\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 29 + \left(42 a + 24\right)\cdot 61 + \left(15 a + 37\right)\cdot 61^{2} + \left(19 a + 49\right)\cdot 61^{3} + \left(10 a + 10\right)\cdot 61^{4} + \left(33 a + 26\right)\cdot 61^{5} + \left(42 a + 30\right)\cdot 61^{6} + \left(40 a + 5\right)\cdot 61^{7} + \left(32 a + 1\right)\cdot 61^{8} + \left(42 a + 22\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 18 + \left(20 a + 28\right)\cdot 61 + \left(39 a + 54\right)\cdot 61^{2} + \left(34 a + 45\right)\cdot 61^{3} + \left(12 a + 12\right)\cdot 61^{4} + \left(24 a + 30\right)\cdot 61^{5} + \left(59 a + 40\right)\cdot 61^{6} + \left(53 a + 53\right)\cdot 61^{7} + \left(43 a + 15\right)\cdot 61^{8} + \left(4 a + 18\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 47\cdot 61 + 22\cdot 61^{2} + 20\cdot 61^{3} + 57\cdot 61^{4} + 50\cdot 61^{5} + 20\cdot 61^{6} + 57\cdot 61^{7} + 53\cdot 61^{8} + 9\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,6,5)$ |
| $(1,4)(2,5)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)(4,5)$ |
$-1$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,2,4)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,2,4)(3,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,6,2,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
