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:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 25 + 45\cdot 61 + \left(60 a + 29\right)\cdot 61^{2} + \left(39 a + 47\right)\cdot 61^{3} + \left(30 a + 54\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 15\cdot 61 + 48\cdot 61^{2} + 55\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 11 + \left(28 a + 54\right)\cdot 61 + \left(57 a + 42\right)\cdot 61^{2} + \left(30 a + 53\right)\cdot 61^{3} + \left(31 a + 20\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 a + 25 + \left(32 a + 7\right)\cdot 61 + \left(3 a + 11\right)\cdot 61^{2} + \left(30 a + 27\right)\cdot 61^{3} + \left(29 a + 21\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 50 + \left(11 a + 24\right)\cdot 61 + \left(44 a + 25\right)\cdot 61^{2} + \left(29 a + 23\right)\cdot 61^{3} + \left(14 a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 a + 18 + \left(49 a + 7\right)\cdot 61 + \left(16 a + 58\right)\cdot 61^{2} + \left(31 a + 8\right)\cdot 61^{3} + \left(46 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 a + 42 + \left(60 a + 28\right)\cdot 61 + 28\cdot 61^{2} + \left(21 a + 27\right)\cdot 61^{3} + \left(30 a + 45\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$-1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
