Basic invariants
| Dimension: | $56$ |
| Group: | $A_8$ |
| Conductor: | \(258\!\cdots\!504\)\(\medspace = 2^{202} \cdot 113^{48} \cdot 911^{48} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.319463173328482073097337827900516204544.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 105 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.319463173328482073097337827900516204544.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823537 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 6 + \left(41 a + 10\right)\cdot 97 + \left(23 a + 12\right)\cdot 97^{2} + \left(79 a + 5\right)\cdot 97^{3} + 23 a\cdot 97^{4} + \left(6 a + 6\right)\cdot 97^{5} + \left(92 a + 73\right)\cdot 97^{6} + \left(65 a + 24\right)\cdot 97^{7} + \left(55 a + 33\right)\cdot 97^{8} + \left(32 a + 6\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 92 a + 35 + \left(49 a + 49\right)\cdot 97 + \left(74 a + 5\right)\cdot 97^{2} + \left(53 a + 94\right)\cdot 97^{3} + \left(42 a + 96\right)\cdot 97^{4} + \left(61 a + 4\right)\cdot 97^{5} + \left(5 a + 86\right)\cdot 97^{6} + \left(49 a + 87\right)\cdot 97^{7} + \left(63 a + 84\right)\cdot 97^{8} + \left(21 a + 41\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 70 a + 22 + \left(3 a + 3\right)\cdot 97 + \left(87 a + 21\right)\cdot 97^{2} + \left(61 a + 63\right)\cdot 97^{3} + \left(65 a + 13\right)\cdot 97^{4} + \left(71 a + 73\right)\cdot 97^{5} + \left(13 a + 44\right)\cdot 97^{6} + \left(46 a + 89\right)\cdot 97^{7} + \left(84 a + 61\right)\cdot 97^{8} + \left(14 a + 89\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 85 a + 54 + \left(55 a + 84\right)\cdot 97 + \left(45 a + 66\right)\cdot 97^{2} + \left(81 a + 8\right)\cdot 97^{3} + \left(10 a + 53\right)\cdot 97^{4} + \left(87 a + 19\right)\cdot 97^{5} + \left(6 a + 44\right)\cdot 97^{6} + \left(45 a + 93\right)\cdot 97^{7} + \left(33 a + 46\right)\cdot 97^{8} + \left(80 a + 51\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 30 + \left(47 a + 7\right)\cdot 97 + \left(22 a + 30\right)\cdot 97^{2} + \left(43 a + 73\right)\cdot 97^{3} + \left(54 a + 85\right)\cdot 97^{4} + \left(35 a + 23\right)\cdot 97^{5} + \left(91 a + 30\right)\cdot 97^{6} + \left(47 a + 34\right)\cdot 97^{7} + \left(33 a + 2\right)\cdot 97^{8} + 75 a\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 92 + \left(93 a + 33\right)\cdot 97 + \left(9 a + 7\right)\cdot 97^{2} + \left(35 a + 38\right)\cdot 97^{3} + \left(31 a + 17\right)\cdot 97^{4} + \left(25 a + 79\right)\cdot 97^{5} + \left(83 a + 83\right)\cdot 97^{6} + \left(50 a + 24\right)\cdot 97^{7} + \left(12 a + 3\right)\cdot 97^{8} + \left(82 a + 20\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a + 42 + \left(41 a + 55\right)\cdot 97 + \left(51 a + 56\right)\cdot 97^{2} + \left(15 a + 44\right)\cdot 97^{3} + \left(86 a + 79\right)\cdot 97^{4} + \left(9 a + 95\right)\cdot 97^{5} + \left(90 a + 60\right)\cdot 97^{6} + \left(51 a + 34\right)\cdot 97^{7} + \left(63 a + 35\right)\cdot 97^{8} + \left(16 a + 1\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 93 a + 10 + \left(55 a + 47\right)\cdot 97 + \left(73 a + 91\right)\cdot 97^{2} + \left(17 a + 60\right)\cdot 97^{3} + \left(73 a + 41\right)\cdot 97^{4} + \left(90 a + 85\right)\cdot 97^{5} + \left(4 a + 61\right)\cdot 97^{6} + \left(31 a + 95\right)\cdot 97^{7} + \left(41 a + 22\right)\cdot 97^{8} + \left(64 a + 80\right)\cdot 97^{9} +O(97^{10})\)
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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 |
| $1$ | $1$ | $()$ | $56$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $-4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.