Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 557 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 557 }$: $ x^{2} + 553 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 216 a + 461 + \left(406 a + 311\right)\cdot 557 + \left(321 a + 201\right)\cdot 557^{2} + \left(135 a + 296\right)\cdot 557^{3} + \left(52 a + 248\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 213 + 459\cdot 557 + 158\cdot 557^{2} + 262\cdot 557^{3} + 111\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 499 + 529\cdot 557 + 104\cdot 557^{2} + 58\cdot 557^{3} + 84\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 342 a + 428 + \left(496 a + 139\right)\cdot 557 + \left(521 a + 543\right)\cdot 557^{2} + \left(326 a + 508\right)\cdot 557^{3} + \left(312 a + 145\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 341 a + 211 + \left(150 a + 50\right)\cdot 557 + \left(235 a + 525\right)\cdot 557^{2} + \left(421 a + 516\right)\cdot 557^{3} + \left(504 a + 321\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 215 a + 125 + \left(60 a + 113\right)\cdot 557 + \left(35 a + 463\right)\cdot 557^{2} + \left(230 a + 180\right)\cdot 557^{3} + \left(244 a + 512\right)\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 293 + 66\cdot 557 + 231\cdot 557^{2} + 404\cdot 557^{3} + 246\cdot 557^{4} +O\left(557^{ 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.
