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:
| $r_{ 1 }$ |
$=$ |
$ 18 a + 14 + \left(24 a + 17\right)\cdot 101 + \left(82 a + 3\right)\cdot 101^{2} + \left(12 a + 63\right)\cdot 101^{3} + \left(69 a + 59\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 74 a + 31 + \left(54 a + 26\right)\cdot 101 + \left(70 a + 80\right)\cdot 101^{2} + \left(16 a + 100\right)\cdot 101^{3} + 24\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 87 + \left(43 a + 68\right)\cdot 101 + \left(17 a + 44\right)\cdot 101^{2} + \left(9 a + 95\right)\cdot 101^{3} + \left(34 a + 91\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 83 a + 86 + \left(76 a + 95\right)\cdot 101 + \left(18 a + 4\right)\cdot 101^{2} + \left(88 a + 32\right)\cdot 101^{3} + \left(31 a + 20\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 88\cdot 101 + 93\cdot 101^{2} + 101^{3} + 80\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 24 + \left(46 a + 70\right)\cdot 101 + \left(30 a + 4\right)\cdot 101^{2} + \left(84 a + 97\right)\cdot 101^{3} + \left(100 a + 8\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 a + 95 + \left(57 a + 36\right)\cdot 101 + \left(83 a + 71\right)\cdot 101^{2} + \left(91 a + 13\right)\cdot 101^{3} + \left(66 a + 17\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$-1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
