Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(135\!\cdots\!856\)\(\medspace = 2^{136} \cdot 11^{39} \cdot 299471^{39} \) |
| Artin stem field: | Galois closure of 8.0.1339937868468943908837290142533567581866950656.1 |
| 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 8.0.1339937868468943908837290142533567581866950656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825792 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 863 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 863 }$:
\( x^{2} + 862x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 848 + 406\cdot 863 + 148\cdot 863^{2} + 731\cdot 863^{3} + 400\cdot 863^{4} + 57\cdot 863^{5} + 590\cdot 863^{6} + 38\cdot 863^{7} + 851\cdot 863^{8} + 150\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 702 a + 276 + \left(854 a + 219\right)\cdot 863 + \left(804 a + 814\right)\cdot 863^{2} + \left(201 a + 229\right)\cdot 863^{3} + \left(856 a + 34\right)\cdot 863^{4} + \left(281 a + 283\right)\cdot 863^{5} + \left(165 a + 92\right)\cdot 863^{6} + \left(630 a + 93\right)\cdot 863^{7} + \left(848 a + 175\right)\cdot 863^{8} + \left(696 a + 476\right)\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 333 + 688\cdot 863 + 160\cdot 863^{2} + 352\cdot 863^{3} + 498\cdot 863^{4} + 438\cdot 863^{5} + 845\cdot 863^{6} + 409\cdot 863^{7} + 681\cdot 863^{8} + 638\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 161 a + 115 + \left(8 a + 372\right)\cdot 863 + \left(58 a + 764\right)\cdot 863^{2} + \left(661 a + 489\right)\cdot 863^{3} + \left(6 a + 688\right)\cdot 863^{4} + \left(581 a + 571\right)\cdot 863^{5} + \left(697 a + 838\right)\cdot 863^{6} + \left(232 a + 557\right)\cdot 863^{7} + \left(14 a + 393\right)\cdot 863^{8} + \left(166 a + 324\right)\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 134 + 666\cdot 863 + 621\cdot 863^{2} + 699\cdot 863^{3} + 517\cdot 863^{4} + 152\cdot 863^{5} + 804\cdot 863^{6} + 857\cdot 863^{7} + 392\cdot 863^{8} + 819\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 205 + 534\cdot 863 + 276\cdot 863^{2} + 300\cdot 863^{3} + 327\cdot 863^{4} + 375\cdot 863^{5} + 120\cdot 863^{6} + 699\cdot 863^{7} + 860\cdot 863^{8} + 286\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 532 a + 73 + \left(541 a + 709\right)\cdot 863 + \left(617 a + 294\right)\cdot 863^{2} + \left(587 a + 339\right)\cdot 863^{3} + \left(457 a + 557\right)\cdot 863^{4} + \left(775 a + 627\right)\cdot 863^{5} + \left(666 a + 134\right)\cdot 863^{6} + \left(278 a + 160\right)\cdot 863^{7} + \left(179 a + 98\right)\cdot 863^{8} + \left(459 a + 669\right)\cdot 863^{9} +O(863^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 331 a + 605 + \left(321 a + 718\right)\cdot 863 + \left(245 a + 370\right)\cdot 863^{2} + \left(275 a + 309\right)\cdot 863^{3} + \left(405 a + 427\right)\cdot 863^{4} + \left(87 a + 82\right)\cdot 863^{5} + \left(196 a + 26\right)\cdot 863^{6} + \left(584 a + 635\right)\cdot 863^{7} + \left(683 a + 861\right)\cdot 863^{8} + \left(403 a + 85\right)\cdot 863^{9} +O(863^{10})\)
|
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.