Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 491 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 491 }$: $ x^{2} + 487 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 292 a + 4 + \left(91 a + 406\right)\cdot 491 + \left(249 a + 160\right)\cdot 491^{2} + \left(305 a + 160\right)\cdot 491^{3} + \left(243 a + 302\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 199 a + 190 + \left(399 a + 480\right)\cdot 491 + \left(241 a + 83\right)\cdot 491^{2} + \left(185 a + 151\right)\cdot 491^{3} + \left(247 a + 480\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 460 a + 14 + \left(391 a + 208\right)\cdot 491 + \left(358 a + 435\right)\cdot 491^{2} + \left(93 a + 232\right)\cdot 491^{3} + \left(389 a + 22\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 130 + \left(293 a + 191\right)\cdot 491 + \left(29 a + 486\right)\cdot 491^{2} + \left(120 a + 31\right)\cdot 491^{3} + \left(249 a + 33\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 479 a + 178 + \left(197 a + 369\right)\cdot 491 + \left(461 a + 311\right)\cdot 491^{2} + \left(370 a + 482\right)\cdot 491^{3} + \left(241 a + 418\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 466\cdot 491 + 479\cdot 491^{2} + 164\cdot 491^{3} + 203\cdot 491^{4} +O\left(491^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 a + 381 + \left(99 a + 333\right)\cdot 491 + \left(132 a + 5\right)\cdot 491^{2} + \left(397 a + 249\right)\cdot 491^{3} + \left(101 a + 12\right)\cdot 491^{4} +O\left(491^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
