Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 313 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 313 }$: $ x^{2} + 310 x + 10 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 195 a + 251 + \left(136 a + 74\right)\cdot 313 + \left(259 a + 150\right)\cdot 313^{2} + \left(146 a + 303\right)\cdot 313^{3} + \left(38 a + 232\right)\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 264 + 13\cdot 313 + 308\cdot 313^{2} + 50\cdot 313^{3} + 44\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 33 + \left(4 a + 93\right)\cdot 313 + \left(17 a + 176\right)\cdot 313^{2} + \left(167 a + 280\right)\cdot 313^{3} + \left(77 a + 10\right)\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 249 a + 283 + \left(151 a + 92\right)\cdot 313 + \left(179 a + 77\right)\cdot 313^{2} + \left(178 a + 288\right)\cdot 313^{3} + \left(110 a + 109\right)\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 118 a + 210 + \left(176 a + 289\right)\cdot 313 + \left(53 a + 165\right)\cdot 313^{2} + \left(166 a + 171\right)\cdot 313^{3} + \left(274 a + 201\right)\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a + 91 + \left(161 a + 299\right)\cdot 313 + \left(133 a + 150\right)\cdot 313^{2} + \left(134 a + 18\right)\cdot 313^{3} + \left(202 a + 263\right)\cdot 313^{4} +O\left(313^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 283 a + 123 + \left(308 a + 75\right)\cdot 313 + \left(295 a + 223\right)\cdot 313^{2} + \left(145 a + 138\right)\cdot 313^{3} + \left(235 a + 76\right)\cdot 313^{4} +O\left(313^{ 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.
