Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 521 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 521 }$: $ x^{2} + 515 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 488 + 5\cdot 521 + 3\cdot 521^{2} + 208\cdot 521^{3} + 195\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 169 a + 518 + \left(14 a + 437\right)\cdot 521 + \left(326 a + 282\right)\cdot 521^{2} + \left(308 a + 342\right)\cdot 521^{3} + \left(289 a + 361\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 155 + 361\cdot 521 + 468\cdot 521^{2} + 423\cdot 521^{3} + 373\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 352 a + 490 + \left(506 a + 354\right)\cdot 521 + \left(194 a + 140\right)\cdot 521^{2} + \left(212 a + 305\right)\cdot 521^{3} + \left(231 a + 227\right)\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 199 + 373\cdot 521 + 481\cdot 521^{2} + 3\cdot 521^{3} + 235\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 327 + 10\cdot 521 + 31\cdot 521^{2} + 419\cdot 521^{3} + 143\cdot 521^{4} +O\left(521^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 428 + 18\cdot 521 + 155\cdot 521^{2} + 381\cdot 521^{3} + 25\cdot 521^{4} +O\left(521^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
