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:
| $r_{ 1 }$ |
$=$ |
$ 55 a + 147 + \left(34 a + 23\right)\cdot 227 + \left(144 a + 79\right)\cdot 227^{2} + \left(72 a + 26\right)\cdot 227^{3} + \left(11 a + 11\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 172 a + 78 + \left(192 a + 208\right)\cdot 227 + \left(82 a + 145\right)\cdot 227^{2} + \left(154 a + 163\right)\cdot 227^{3} + \left(215 a + 17\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 75\cdot 227 + 6\cdot 227^{2} + 93\cdot 227^{3} + 121\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 69 + 19\cdot 227 + 162\cdot 227^{2} + 90\cdot 227^{3} + 180\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 72 + \left(98 a + 80\right)\cdot 227 + \left(64 a + 162\right)\cdot 227^{2} + \left(65 a + 25\right)\cdot 227^{3} + \left(34 a + 70\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 224 + 219\cdot 227 + 63\cdot 227^{2} + 90\cdot 227^{3} + 35\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 195 a + 69 + \left(128 a + 54\right)\cdot 227 + \left(162 a + 61\right)\cdot 227^{2} + \left(161 a + 191\right)\cdot 227^{3} + \left(192 a + 17\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 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.
