Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 257 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 257 }$: $ x^{2} + 251 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 237 + 69\cdot 257 + 255\cdot 257^{2} + 122\cdot 257^{3} + 199\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 55 + \left(31 a + 67\right)\cdot 257 + \left(95 a + 88\right)\cdot 257^{2} + \left(77 a + 154\right)\cdot 257^{3} + \left(188 a + 45\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 157 + 45\cdot 257 + 116\cdot 257^{2} + 2\cdot 257^{3} + 46\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 251 a + 91 + \left(225 a + 247\right)\cdot 257 + \left(161 a + 113\right)\cdot 257^{2} + \left(179 a + 9\right)\cdot 257^{3} + \left(68 a + 70\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 214 a + 132 + \left(209 a + 113\right)\cdot 257 + \left(209 a + 193\right)\cdot 257^{2} + \left(150 a + 197\right)\cdot 257^{3} + \left(72 a + 243\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 225 + 96\cdot 257 + 46\cdot 257^{2} + 162\cdot 257^{3} + 151\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 a + 131 + \left(47 a + 130\right)\cdot 257 + \left(47 a + 214\right)\cdot 257^{2} + \left(106 a + 121\right)\cdot 257^{3} + \left(184 a + 14\right)\cdot 257^{4} +O\left(257^{ 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.
