Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: $ x^{2} + 220 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 186 + 196\cdot 227 + 157\cdot 227^{2} + 67\cdot 227^{3} + 104\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 a + 7 + \left(81 a + 189\right)\cdot 227 + \left(209 a + 221\right)\cdot 227^{2} + 72\cdot 227^{3} + \left(129 a + 197\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 156 a + 50 + \left(145 a + 6\right)\cdot 227 + \left(17 a + 17\right)\cdot 227^{2} + \left(226 a + 97\right)\cdot 227^{3} + \left(97 a + 191\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 91 a + 35 + \left(184 a + 196\right)\cdot 227 + \left(24 a + 56\right)\cdot 227^{2} + \left(216 a + 214\right)\cdot 227^{3} + \left(211 a + 151\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 223\cdot 227 + 168\cdot 227^{2} + 50\cdot 227^{3} + 206\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 158 + 62\cdot 227 + 12\cdot 227^{2} + 65\cdot 227^{3} + 226\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 136 a + 218 + \left(42 a + 33\right)\cdot 227 + \left(202 a + 46\right)\cdot 227^{2} + \left(10 a + 113\right)\cdot 227^{3} + \left(15 a + 57\right)\cdot 227^{4} +O\left(227^{ 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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
