Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 499 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 499 }$: $ x^{2} + 493 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 + 161\cdot 499 + 74\cdot 499^{2} + 499^{3} + 296\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 466 + 437\cdot 499 + 496\cdot 499^{2} + 488\cdot 499^{3} + 207\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 134 + 379\cdot 499 + 421\cdot 499^{2} + 15\cdot 499^{3} + 415\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 211 + 87\cdot 499 + 98\cdot 499^{2} + 353\cdot 499^{3} + 355\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 400 a + 481 + \left(324 a + 448\right)\cdot 499 + \left(456 a + 437\right)\cdot 499^{2} + \left(222 a + 270\right)\cdot 499^{3} + \left(449 a + 3\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 a + 386 + \left(174 a + 1\right)\cdot 499 + \left(42 a + 358\right)\cdot 499^{2} + \left(276 a + 153\right)\cdot 499^{3} + \left(49 a + 481\right)\cdot 499^{4} +O\left(499^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 276 + 479\cdot 499 + 108\cdot 499^{2} + 213\cdot 499^{3} + 236\cdot 499^{4} +O\left(499^{ 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.
