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:
| $r_{ 1 }$ |
$=$ |
$ 128 + 168\cdot 227 + 137\cdot 227^{2} + 129\cdot 227^{3} + 178\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 149\cdot 227 + 170\cdot 227^{2} + 56\cdot 227^{3} + 186\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 157\cdot 227 + 55\cdot 227^{2} + 95\cdot 227^{3} + 2\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 a + 157 + \left(121 a + 116\right)\cdot 227 + \left(167 a + 170\right)\cdot 227^{2} + \left(3 a + 69\right)\cdot 227^{3} + \left(33 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 150 a + 15 + \left(105 a + 208\right)\cdot 227 + \left(59 a + 86\right)\cdot 227^{2} + \left(223 a + 155\right)\cdot 227^{3} + \left(193 a + 29\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 190 a + 59 + \left(137 a + 7\right)\cdot 227 + \left(131 a + 92\right)\cdot 227^{2} + \left(194 a + 39\right)\cdot 227^{3} + \left(133 a + 210\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 a + 27 + \left(89 a + 101\right)\cdot 227 + \left(95 a + 194\right)\cdot 227^{2} + \left(32 a + 134\right)\cdot 227^{3} + \left(93 a + 44\right)\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(3,4,5,6,7)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $360$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $360$ | $7$ | $(1,3,4,5,6,7,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
