Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(193\!\cdots\!128\)\(\medspace = 2^{130} \cdot 29^{39} \cdot 37^{39} \cdot 49121^{39} \) |
| Artin stem field: | Galois closure of 8.0.5620066455663197695561554590870531303687507769819136.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 336 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.5620066455663197695561554590870531303687507769819136.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825540 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 823 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 823 }$:
\( x^{2} + 821x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 163 a + 122 + \left(660 a + 505\right)\cdot 823 + \left(645 a + 316\right)\cdot 823^{2} + \left(568 a + 618\right)\cdot 823^{3} + \left(209 a + 397\right)\cdot 823^{4} + \left(753 a + 137\right)\cdot 823^{5} + \left(227 a + 224\right)\cdot 823^{6} + \left(686 a + 243\right)\cdot 823^{7} + \left(229 a + 456\right)\cdot 823^{8} + \left(544 a + 45\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 472 a + 166 + \left(539 a + 211\right)\cdot 823 + \left(357 a + 652\right)\cdot 823^{2} + \left(657 a + 795\right)\cdot 823^{3} + \left(53 a + 65\right)\cdot 823^{4} + \left(205 a + 6\right)\cdot 823^{5} + \left(808 a + 84\right)\cdot 823^{6} + \left(576 a + 123\right)\cdot 823^{7} + \left(65 a + 346\right)\cdot 823^{8} + \left(615 a + 431\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 351 a + 287 + \left(283 a + 818\right)\cdot 823 + \left(465 a + 4\right)\cdot 823^{2} + \left(165 a + 107\right)\cdot 823^{3} + \left(769 a + 339\right)\cdot 823^{4} + \left(617 a + 362\right)\cdot 823^{5} + \left(14 a + 672\right)\cdot 823^{6} + \left(246 a + 468\right)\cdot 823^{7} + \left(757 a + 723\right)\cdot 823^{8} + \left(207 a + 772\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 a + 91 + \left(633 a + 277\right)\cdot 823 + \left(44 a + 259\right)\cdot 823^{2} + \left(565 a + 344\right)\cdot 823^{3} + \left(102 a + 86\right)\cdot 823^{4} + \left(626 a + 680\right)\cdot 823^{5} + \left(294 a + 351\right)\cdot 823^{6} + \left(455 a + 486\right)\cdot 823^{7} + \left(413 a + 454\right)\cdot 823^{8} + \left(301 a + 556\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 652 a + 741 + \left(799 a + 751\right)\cdot 823 + \left(117 a + 567\right)\cdot 823^{2} + \left(69 a + 667\right)\cdot 823^{3} + \left(572 a + 264\right)\cdot 823^{4} + \left(187 a + 342\right)\cdot 823^{5} + \left(571 a + 381\right)\cdot 823^{6} + \left(329 a + 107\right)\cdot 823^{7} + \left(349 a + 766\right)\cdot 823^{8} + \left(527 a + 387\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 660 a + 448 + \left(162 a + 16\right)\cdot 823 + \left(177 a + 125\right)\cdot 823^{2} + \left(254 a + 287\right)\cdot 823^{3} + \left(613 a + 248\right)\cdot 823^{4} + \left(69 a + 611\right)\cdot 823^{5} + \left(595 a + 749\right)\cdot 823^{6} + \left(136 a + 564\right)\cdot 823^{7} + \left(593 a + 229\right)\cdot 823^{8} + \left(278 a + 81\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 761 a + 215 + \left(189 a + 658\right)\cdot 823 + \left(778 a + 538\right)\cdot 823^{2} + \left(257 a + 606\right)\cdot 823^{3} + \left(720 a + 549\right)\cdot 823^{4} + \left(196 a + 183\right)\cdot 823^{5} + \left(528 a + 315\right)\cdot 823^{6} + \left(367 a + 279\right)\cdot 823^{7} + \left(409 a + 3\right)\cdot 823^{8} + \left(521 a + 746\right)\cdot 823^{9} +O(823^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 171 a + 399 + \left(23 a + 53\right)\cdot 823 + \left(705 a + 4\right)\cdot 823^{2} + \left(753 a + 688\right)\cdot 823^{3} + \left(250 a + 516\right)\cdot 823^{4} + \left(635 a + 145\right)\cdot 823^{5} + \left(251 a + 513\right)\cdot 823^{6} + \left(493 a + 195\right)\cdot 823^{7} + \left(473 a + 312\right)\cdot 823^{8} + \left(295 a + 270\right)\cdot 823^{9} +O(823^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $45$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-3$ |
| $112$ | $3$ | $(1,2,3)$ | $0$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.