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 value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
