Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 617 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 603\cdot 617 + 352\cdot 617^{2} + 6\cdot 617^{3} + 227\cdot 617^{4} + 126\cdot 617^{5} + 427\cdot 617^{6} + 464\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 112 + 426\cdot 617 + 571\cdot 617^{2} + 28\cdot 617^{3} + 340\cdot 617^{4} + 262\cdot 617^{5} + 14\cdot 617^{6} + 512\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 147 + 176\cdot 617 + 432\cdot 617^{2} + 217\cdot 617^{3} + 424\cdot 617^{4} + 112\cdot 617^{5} + 497\cdot 617^{6} + 349\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 174 + 459\cdot 617 + 269\cdot 617^{2} + 199\cdot 617^{3} + 594\cdot 617^{4} + 528\cdot 617^{5} + 539\cdot 617^{6} + 156\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 184 + 172\cdot 617 + 577\cdot 617^{2} + 170\cdot 617^{3} + 492\cdot 617^{4} + 329\cdot 617^{5} + 182\cdot 617^{6} + 215\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 318 + 32\cdot 617 + 349\cdot 617^{2} + 410\cdot 617^{3} + 174\cdot 617^{4} + 515\cdot 617^{5} + 609\cdot 617^{6} + 41\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 345 + 499\cdot 617 + 58\cdot 617^{2} + 199\cdot 617^{3} + 437\cdot 617^{4} + 561\cdot 617^{5} + 7\cdot 617^{6} + 226\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 568 + 98\cdot 617 + 473\cdot 617^{2} + 395\cdot 617^{4} + 30\cdot 617^{5} + 189\cdot 617^{6} + 501\cdot 617^{7} +O\left(617^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(3,4)$ |
| $(1,8,4,5,6,7,3,2)$ |
| $(1,3,6,4)(2,8,5,7)$ |
| $(1,6)(7,8)$ |
| $(2,5)(7,8)$ |
| $(1,2,6,5)(3,7,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-4$ |
| $2$ | $2$ | $(2,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(2,5)(3,4)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $8$ | $2$ | $(2,7)(3,4)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,6,3)(2,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,8,5,7)$ | $0$ |
| $4$ | $4$ | $(1,2,6,5)(3,7,4,8)$ | $0$ |
| $4$ | $4$ | $(1,5,6,2)(3,7,4,8)$ | $0$ |
| $4$ | $4$ | $(2,7,5,8)$ | $-2$ |
| $4$ | $4$ | $(1,4,6,3)(2,5)(7,8)$ | $2$ |
| $8$ | $8$ | $(1,8,4,5,6,7,3,2)$ | $0$ |
| $8$ | $8$ | $(1,8,4,2,6,7,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
