Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 219 a + 56 + \left(197 a + 56\right)\cdot 277 + \left(158 a + 117\right)\cdot 277^{2} + \left(201 a + 38\right)\cdot 277^{3} + \left(203 a + 196\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 152 + 202\cdot 277 + 96\cdot 277^{2} + 256\cdot 277^{3} + 43\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 222 + 213\cdot 277 + 231\cdot 277^{2} + 36\cdot 277^{3} + 253\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 273 + 88\cdot 277 + 172\cdot 277^{2} + 89\cdot 277^{3} + 191\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 271 a + 132 + \left(276 a + 193\right)\cdot 277 + \left(48 a + 250\right)\cdot 277^{2} + \left(103 a + 247\right)\cdot 277^{3} + \left(250 a + 138\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 159 + \left(79 a + 153\right)\cdot 277 + \left(118 a + 118\right)\cdot 277^{2} + \left(75 a + 207\right)\cdot 277^{3} + \left(73 a + 51\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 a + 114 + 199\cdot 277 + \left(228 a + 120\right)\cdot 277^{2} + \left(173 a + 231\right)\cdot 277^{3} + \left(26 a + 232\right)\cdot 277^{4} +O\left(277^{ 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.
