Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 353 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 353 }$: $ x^{2} + 348 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 232 a + 227 + \left(27 a + 2\right)\cdot 353 + \left(298 a + 81\right)\cdot 353^{2} + \left(349 a + 121\right)\cdot 353^{3} + \left(152 a + 338\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 118 a + 151 + \left(107 a + 162\right)\cdot 353 + \left(302 a + 132\right)\cdot 353^{2} + \left(134 a + 105\right)\cdot 353^{3} + \left(84 a + 279\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 243 + 159\cdot 353 + 14\cdot 353^{2} + 247\cdot 353^{3} + 192\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 121 a + 328 + \left(325 a + 261\right)\cdot 353 + \left(54 a + 131\right)\cdot 353^{2} + \left(3 a + 160\right)\cdot 353^{3} + \left(200 a + 47\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 130 + 229\cdot 353 + 177\cdot 353^{2} + 315\cdot 353^{3} + 193\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 298 + 14\cdot 353 + 44\cdot 353^{2} + 338\cdot 353^{3} + 146\cdot 353^{4} +O\left(353^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 235 a + 35 + \left(245 a + 228\right)\cdot 353 + \left(50 a + 124\right)\cdot 353^{2} + \left(218 a + 124\right)\cdot 353^{3} + \left(268 a + 213\right)\cdot 353^{4} +O\left(353^{ 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.
