Basic invariants
| Dimension: | $21$ |
| Group: | $A_8$ |
| Conductor: | \(896\!\cdots\!384\)\(\medspace = 2^{68} \cdot 823547^{18} \) |
| Artin number field: | Galois closure of 8.0.81784359890073480322911752930604437733376.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 336 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.81784359890073480322911752930604437733376.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 941 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 941 }$:
\( x^{2} + 940x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 145 + 701\cdot 941 + 272\cdot 941^{2} + 288\cdot 941^{3} + 311\cdot 941^{4} + 585\cdot 941^{5} + 28\cdot 941^{6} + 15\cdot 941^{7} + 839\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 440 a + 67 + \left(881 a + 158\right)\cdot 941 + \left(823 a + 515\right)\cdot 941^{2} + \left(600 a + 639\right)\cdot 941^{3} + \left(743 a + 412\right)\cdot 941^{4} + \left(452 a + 442\right)\cdot 941^{5} + \left(323 a + 388\right)\cdot 941^{6} + \left(107 a + 222\right)\cdot 941^{7} + \left(716 a + 796\right)\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 741 a + 681 + \left(404 a + 692\right)\cdot 941 + \left(822 a + 466\right)\cdot 941^{2} + \left(889 a + 409\right)\cdot 941^{3} + \left(276 a + 610\right)\cdot 941^{4} + \left(474 a + 686\right)\cdot 941^{5} + \left(752 a + 55\right)\cdot 941^{6} + \left(201 a + 86\right)\cdot 941^{7} + \left(781 a + 801\right)\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 200 a + 481 + \left(536 a + 356\right)\cdot 941 + \left(118 a + 884\right)\cdot 941^{2} + \left(51 a + 476\right)\cdot 941^{3} + \left(664 a + 938\right)\cdot 941^{4} + \left(466 a + 883\right)\cdot 941^{5} + \left(188 a + 333\right)\cdot 941^{6} + \left(739 a + 476\right)\cdot 941^{7} + \left(159 a + 439\right)\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 501 a + 507 + \left(59 a + 599\right)\cdot 941 + \left(117 a + 457\right)\cdot 941^{2} + \left(340 a + 416\right)\cdot 941^{3} + \left(197 a + 555\right)\cdot 941^{4} + \left(488 a + 151\right)\cdot 941^{5} + \left(617 a + 259\right)\cdot 941^{6} + \left(833 a + 6\right)\cdot 941^{7} + \left(224 a + 464\right)\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 880 + 6\cdot 941 + 163\cdot 941^{2} + 642\cdot 941^{3} + 516\cdot 941^{4} + 229\cdot 941^{5} + 908\cdot 941^{6} + 511\cdot 941^{7} + 863\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 431 + 716\cdot 941 + 530\cdot 941^{2} + 401\cdot 941^{3} + 880\cdot 941^{4} + 124\cdot 941^{5} + 826\cdot 941^{6} + 279\cdot 941^{7} + 22\cdot 941^{8} +O(941^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 572 + 532\cdot 941 + 473\cdot 941^{2} + 489\cdot 941^{3} + 479\cdot 941^{4} + 659\cdot 941^{5} + 22\cdot 941^{6} + 284\cdot 941^{7} + 479\cdot 941^{8} +O(941^{9})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $21$ | $21$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $1$ | $1$ |
| $112$ | $3$ | $(1,2,3)$ | $-3$ | $-3$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ | $-1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ | $1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} + 2 \zeta_{15} - 2$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - 2 \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - 2 \zeta_{15} + 1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - 2 \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} - 2 \zeta_{15} + 1$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} + 2 \zeta_{15} - 2$ |