Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 433 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 433 }$: $ x^{2} + 432 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 193 + 119\cdot 433 + 23\cdot 433^{2} + 54\cdot 433^{3} + 159\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 204\cdot 433 + 9\cdot 433^{2} + 46\cdot 433^{3} + 400\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 294 a + 23 + \left(290 a + 234\right)\cdot 433 + \left(68 a + 158\right)\cdot 433^{2} + \left(405 a + 189\right)\cdot 433^{3} + \left(342 a + 204\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 148 + 171\cdot 433 + 107\cdot 433^{2} + 75\cdot 433^{3} + 245\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 139 a + 317 + \left(142 a + 230\right)\cdot 433 + \left(364 a + 369\right)\cdot 433^{2} + \left(27 a + 92\right)\cdot 433^{3} + \left(90 a + 142\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 206 a + 406 + \left(196 a + 390\right)\cdot 433 + \left(426 a + 416\right)\cdot 433^{2} + \left(223 a + 88\right)\cdot 433^{3} + \left(392 a + 206\right)\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 227 a + 179 + \left(236 a + 381\right)\cdot 433 + \left(6 a + 213\right)\cdot 433^{2} + \left(209 a + 319\right)\cdot 433^{3} + \left(40 a + 374\right)\cdot 433^{4} +O\left(433^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
