Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 283 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 283 }$: $ x^{2} + 282 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 260 + \left(202 a + 280\right)\cdot 283 + \left(238 a + 178\right)\cdot 283^{2} + \left(267 a + 124\right)\cdot 283^{3} + \left(54 a + 160\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 a + 113 + \left(181 a + 98\right)\cdot 283 + \left(241 a + 41\right)\cdot 283^{2} + \left(111 a + 43\right)\cdot 283^{3} + \left(117 a + 119\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 143 + 255\cdot 283 + 228\cdot 283^{2} + 67\cdot 283^{3} + 157\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 + 16\cdot 283 + 32\cdot 283^{2} + 5\cdot 283^{3} + 140\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 240 a + 20 + \left(80 a + 157\right)\cdot 283 + \left(44 a + 215\right)\cdot 283^{2} + \left(15 a + 153\right)\cdot 283^{3} + \left(228 a + 230\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 214 a + 182 + \left(101 a + 210\right)\cdot 283 + \left(41 a + 101\right)\cdot 283^{2} + \left(171 a + 196\right)\cdot 283^{3} + \left(165 a + 124\right)\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 113\cdot 283 + 50\cdot 283^{2} + 258\cdot 283^{3} + 199\cdot 283^{4} +O\left(283^{ 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.
