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:
| $r_{ 1 }$ |
$=$ |
$ 55 + 149\cdot 277 + 132\cdot 277^{2} + 244\cdot 277^{3} + 22\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 187 + 136\cdot 277 + 30\cdot 277^{2} + 274\cdot 277^{3} + 2\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 153 a + 49 + \left(9 a + 11\right)\cdot 277 + \left(254 a + 207\right)\cdot 277^{2} + \left(104 a + 234\right)\cdot 277^{3} + \left(223 a + 234\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 150 a + 27 + \left(207 a + 159\right)\cdot 277 + \left(182 a + 222\right)\cdot 277^{2} + \left(90 a + 101\right)\cdot 277^{3} + \left(221 a + 247\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 127 a + 200 + \left(69 a + 77\right)\cdot 277 + \left(94 a + 9\right)\cdot 277^{2} + \left(186 a + 191\right)\cdot 277^{3} + \left(55 a + 266\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 124 a + 231 + \left(267 a + 163\right)\cdot 277 + \left(22 a + 128\right)\cdot 277^{2} + \left(172 a + 18\right)\cdot 277^{3} + \left(53 a + 246\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 133\cdot 277 + 100\cdot 277^{2} + 43\cdot 277^{3} + 87\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
