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:
| $r_{ 1 }$ |
$=$ |
$ 241 a + 232 + \left(229 a + 4\right)\cdot 257 + \left(97 a + 203\right)\cdot 257^{2} + \left(200 a + 15\right)\cdot 257^{3} + \left(113 a + 76\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 252 a + 116 + \left(124 a + 194\right)\cdot 257 + \left(86 a + 130\right)\cdot 257^{2} + \left(214 a + 136\right)\cdot 257^{3} + \left(34 a + 118\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 136 + \left(27 a + 115\right)\cdot 257 + \left(159 a + 46\right)\cdot 257^{2} + \left(56 a + 92\right)\cdot 257^{3} + \left(143 a + 44\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 86 + \left(132 a + 178\right)\cdot 257 + \left(170 a + 10\right)\cdot 257^{2} + \left(42 a + 51\right)\cdot 257^{3} + \left(222 a + 113\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 71 + 162\cdot 257 + 110\cdot 257^{2} + 45\cdot 257^{3} + 35\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 212 a + 200 + \left(155 a + 81\right)\cdot 257 + \left(184 a + 44\right)\cdot 257^{2} + \left(221 a + 156\right)\cdot 257^{3} + \left(176 a + 157\right)\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 a + 187 + \left(101 a + 33\right)\cdot 257 + \left(72 a + 225\right)\cdot 257^{2} + \left(35 a + 16\right)\cdot 257^{3} + \left(80 a + 226\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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
