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:
| $r_{ 1 }$ |
$=$ |
$ 168 a + 90 + \left(208 a + 2\right)\cdot 277 + \left(236 a + 96\right)\cdot 277^{2} + \left(152 a + 64\right)\cdot 277^{3} + \left(129 a + 87\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 177 a + 107 + \left(86 a + 169\right)\cdot 277 + \left(32 a + 270\right)\cdot 277^{2} + \left(165 a + 183\right)\cdot 277^{3} + \left(63 a + 168\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 167 a + 114 + \left(265 a + 45\right)\cdot 277 + \left(177 a + 249\right)\cdot 277^{2} + \left(125 a + 27\right)\cdot 277^{3} + \left(57 a + 75\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 100 a + 84 + \left(190 a + 252\right)\cdot 277 + \left(244 a + 3\right)\cdot 277^{2} + \left(111 a + 93\right)\cdot 277^{3} + \left(213 a + 194\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 + 57\cdot 277 + 204\cdot 277^{2} + 225\cdot 277^{3} + 137\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 109 a + 40 + \left(68 a + 183\right)\cdot 277 + \left(40 a + 43\right)\cdot 277^{2} + \left(124 a + 9\right)\cdot 277^{3} + \left(147 a + 46\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 110 a + 61 + \left(11 a + 121\right)\cdot 277 + \left(99 a + 240\right)\cdot 277^{2} + \left(151 a + 226\right)\cdot 277^{3} + \left(219 a + 121\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
