Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: $ x^{2} + 221 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 57 + \left(101 a + 4\right)\cdot 223 + 98\cdot 223^{2} + \left(91 a + 34\right)\cdot 223^{3} + \left(125 a + 89\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 174 a + 59 + \left(86 a + 158\right)\cdot 223 + \left(116 a + 140\right)\cdot 223^{2} + \left(25 a + 36\right)\cdot 223^{3} + \left(33 a + 162\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 83 + 56\cdot 223 + 204\cdot 223^{2} + 218\cdot 223^{3} + 116\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 184 + \left(136 a + 157\right)\cdot 223 + \left(106 a + 63\right)\cdot 223^{2} + \left(197 a + 194\right)\cdot 223^{3} + \left(189 a + 202\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 213 a + 77 + \left(121 a + 196\right)\cdot 223 + \left(222 a + 220\right)\cdot 223^{2} + \left(131 a + 215\right)\cdot 223^{3} + \left(97 a + 25\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 155 + 129\cdot 223 + 145\cdot 223^{2} + 45\cdot 223^{3} + 136\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 189\cdot 223 + 18\cdot 223^{2} + 146\cdot 223^{3} + 158\cdot 223^{4} +O\left(223^{ 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 value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
