Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 39 + \left(69 a + 61\right)\cdot 101 + \left(36 a + 2\right)\cdot 101^{2} + \left(35 a + 48\right)\cdot 101^{3} + \left(13 a + 73\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 87 a + 70 + \left(91 a + 9\right)\cdot 101 + \left(46 a + 40\right)\cdot 101^{2} + \left(76 a + 51\right)\cdot 101^{3} + \left(67 a + 84\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 33\cdot 101 + 58\cdot 101^{2} + 33\cdot 101^{3} + 70\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 80 a + 22 + \left(31 a + 14\right)\cdot 101 + \left(64 a + 80\right)\cdot 101^{2} + \left(65 a + 51\right)\cdot 101^{3} + \left(87 a + 91\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 21 + \left(70 a + 29\right)\cdot 101 + \left(88 a + 2\right)\cdot 101^{2} + \left(92 a + 15\right)\cdot 101^{3} + \left(43 a + 63\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 14 + \left(9 a + 88\right)\cdot 101 + \left(54 a + 34\right)\cdot 101^{2} + \left(24 a + 7\right)\cdot 101^{3} + \left(33 a + 77\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 a + 88 + \left(30 a + 66\right)\cdot 101 + \left(12 a + 84\right)\cdot 101^{2} + \left(8 a + 95\right)\cdot 101^{3} + \left(57 a + 44\right)\cdot 101^{4} +O\left(101^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$35$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
