Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 409 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 409 }$: $ x^{2} + 404 x + 21 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 165 a + 307 + \left(114 a + 219\right)\cdot 409 + \left(408 a + 168\right)\cdot 409^{2} + \left(60 a + 384\right)\cdot 409^{3} + \left(327 a + 215\right)\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 244 a + 314 + \left(294 a + 217\right)\cdot 409 + 50\cdot 409^{2} + \left(348 a + 281\right)\cdot 409^{3} + \left(81 a + 154\right)\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 152 a + 191 + \left(382 a + 395\right)\cdot 409 + \left(3 a + 291\right)\cdot 409^{2} + \left(146 a + 7\right)\cdot 409^{3} + \left(10 a + 258\right)\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 167 + 2\cdot 409 + 25\cdot 409^{2} + 280\cdot 409^{3} + 12\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 358 a + 186 + \left(16 a + 277\right)\cdot 409 + \left(280 a + 302\right)\cdot 409^{2} + \left(307 a + 367\right)\cdot 409^{3} + \left(140 a + 12\right)\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 257 a + 133 + \left(26 a + 110\right)\cdot 409 + \left(405 a + 338\right)\cdot 409^{2} + \left(262 a + 324\right)\cdot 409^{3} + \left(398 a + 163\right)\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 a + 340 + \left(392 a + 3\right)\cdot 409 + \left(128 a + 50\right)\cdot 409^{2} + \left(101 a + 399\right)\cdot 409^{3} + \left(268 a + 408\right)\cdot 409^{4} +O\left(409^{ 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 value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
