Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 349 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 349 }$: $ x^{2} + 348 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 168 a + 19 + \left(298 a + 120\right)\cdot 349 + \left(113 a + 56\right)\cdot 349^{2} + \left(222 a + 96\right)\cdot 349^{3} + \left(138 a + 117\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 181 a + 187 + \left(50 a + 250\right)\cdot 349 + \left(235 a + 220\right)\cdot 349^{2} + \left(126 a + 204\right)\cdot 349^{3} + \left(210 a + 33\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 318 a + 211 + \left(57 a + 110\right)\cdot 349 + \left(136 a + 207\right)\cdot 349^{2} + \left(265 a + 114\right)\cdot 349^{3} + \left(19 a + 317\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 158 a + 284 + \left(283 a + 227\right)\cdot 349 + \left(141 a + 49\right)\cdot 349^{2} + \left(33 a + 9\right)\cdot 349^{3} + \left(208 a + 48\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 180 + \left(291 a + 199\right)\cdot 349 + \left(212 a + 285\right)\cdot 349^{2} + \left(83 a + 243\right)\cdot 349^{3} + \left(329 a + 71\right)\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 + 134\cdot 349 + 319\cdot 349^{2} + 128\cdot 349^{3} + 236\cdot 349^{4} +O\left(349^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 191 a + 93 + \left(65 a + 4\right)\cdot 349 + \left(207 a + 257\right)\cdot 349^{2} + \left(315 a + 249\right)\cdot 349^{3} + \left(140 a + 222\right)\cdot 349^{4} +O\left(349^{ 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.
