Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 503 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 503 }$: $ x^{2} + 498 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 + 315\cdot 503 + 173\cdot 503^{2} + 434\cdot 503^{3} + 350\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 191 a + 312 + \left(355 a + 370\right)\cdot 503 + \left(160 a + 383\right)\cdot 503^{2} + \left(127 a + 261\right)\cdot 503^{3} + \left(410 a + 84\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 312 a + 261 + \left(147 a + 447\right)\cdot 503 + \left(342 a + 328\right)\cdot 503^{2} + \left(375 a + 234\right)\cdot 503^{3} + \left(92 a + 499\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 461 + 246\cdot 503 + 411\cdot 503^{2} + 410\cdot 503^{3} + 20\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 210 a + 130 + \left(452 a + 464\right)\cdot 503 + \left(81 a + 428\right)\cdot 503^{2} + \left(68 a + 299\right)\cdot 503^{3} + \left(81 a + 307\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 293 a + 174 + \left(50 a + 1\right)\cdot 503 + \left(421 a + 386\right)\cdot 503^{2} + \left(434 a + 55\right)\cdot 503^{3} + \left(421 a + 142\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 137 + 166\cdot 503 + 402\cdot 503^{2} + 314\cdot 503^{3} + 103\cdot 503^{4} +O\left(503^{ 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.
