Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 337 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 337 }$: $ x^{2} + 332 x + 10 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 217 a + 224 + \left(256 a + 128\right)\cdot 337 + \left(161 a + 17\right)\cdot 337^{2} + \left(292 a + 202\right)\cdot 337^{3} + \left(157 a + 241\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 120 a + 298 + \left(80 a + 183\right)\cdot 337 + \left(175 a + 232\right)\cdot 337^{2} + \left(44 a + 154\right)\cdot 337^{3} + \left(179 a + 64\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 279 a + 85 + \left(108 a + 189\right)\cdot 337 + \left(306 a + 130\right)\cdot 337^{2} + \left(126 a + 190\right)\cdot 337^{3} + \left(131 a + 335\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 a + 132 + \left(228 a + 117\right)\cdot 337 + \left(30 a + 205\right)\cdot 337^{2} + \left(210 a + 181\right)\cdot 337^{3} + \left(205 a + 191\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 271 + 149\cdot 337 + 169\cdot 337^{2} + 84\cdot 337^{3} + 294\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 256 + \left(265 a + 148\right)\cdot 337 + \left(137 a + 84\right)\cdot 337^{2} + \left(138 a + 327\right)\cdot 337^{3} + \left(50 a + 221\right)\cdot 337^{4} +O\left(337^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 304 a + 84 + \left(71 a + 93\right)\cdot 337 + \left(199 a + 171\right)\cdot 337^{2} + \left(198 a + 207\right)\cdot 337^{3} + \left(286 a + 335\right)\cdot 337^{4} +O\left(337^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$35$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
