Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 311 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$: $ x^{2} + 310 x + 17 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 59 + \left(122 a + 217\right)\cdot 311 + \left(123 a + 254\right)\cdot 311^{2} + \left(310 a + 253\right)\cdot 311^{3} + \left(29 a + 44\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 262 a + 108 + \left(188 a + 290\right)\cdot 311 + \left(187 a + 255\right)\cdot 311^{2} + 129\cdot 311^{3} + \left(281 a + 75\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 218 a + 271 + \left(261 a + 66\right)\cdot 311 + \left(163 a + 90\right)\cdot 311^{2} + \left(74 a + 82\right)\cdot 311^{3} + \left(17 a + 273\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 34\cdot 311 + 94\cdot 311^{2} + 169\cdot 311^{3} + 85\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 a + 178 + \left(49 a + 110\right)\cdot 311 + \left(147 a + 303\right)\cdot 311^{2} + \left(236 a + 303\right)\cdot 311^{3} + \left(293 a + 215\right)\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 261 + 263\cdot 311 + 79\cdot 311^{2} + 169\cdot 311^{3} + 9\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 304 + 260\cdot 311 + 165\cdot 311^{2} + 135\cdot 311^{3} + 228\cdot 311^{4} +O\left(311^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
