Basic invariants
| Dimension: | $28$ |
| Group: | $A_8$ |
| Conductor: | \(735\!\cdots\!664\)\(\medspace = 2^{86} \cdot 823643^{24} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.81841577658693500678182700989203572064256.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 56 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.81841577658693500678182700989203572064256.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823636 \)
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The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 75 a + 29 + \left(78 a + 87\right)\cdot 109 + \left(23 a + 76\right)\cdot 109^{2} + \left(37 a + 43\right)\cdot 109^{3} + \left(54 a + 27\right)\cdot 109^{4} + \left(87 a + 99\right)\cdot 109^{5} + \left(44 a + 60\right)\cdot 109^{6} + \left(75 a + 43\right)\cdot 109^{7} + \left(60 a + 75\right)\cdot 109^{8} + \left(102 a + 34\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 108 a + 107 + \left(15 a + 80\right)\cdot 109 + \left(34 a + 15\right)\cdot 109^{2} + \left(93 a + 80\right)\cdot 109^{3} + \left(33 a + 54\right)\cdot 109^{4} + \left(105 a + 68\right)\cdot 109^{5} + \left(15 a + 65\right)\cdot 109^{6} + \left(54 a + 48\right)\cdot 109^{7} + \left(37 a + 67\right)\cdot 109^{8} + 84 a\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 13\cdot 109 + 45\cdot 109^{2} + 44\cdot 109^{3} + 42\cdot 109^{4} + 28\cdot 109^{5} + 32\cdot 109^{6} + 40\cdot 109^{7} + 24\cdot 109^{8} + 41\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 93 + 83\cdot 109^{2} + 99\cdot 109^{3} + 28\cdot 109^{4} + 97\cdot 109^{5} + 78\cdot 109^{6} + 70\cdot 109^{7} + 59\cdot 109^{8} + 85\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 99 + 13\cdot 109 + 6\cdot 109^{2} + 103\cdot 109^{3} + 97\cdot 109^{4} + 84\cdot 109^{5} + 76\cdot 109^{6} + 93\cdot 109^{7} + 94\cdot 109^{8} + 13\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 a + 104 + \left(30 a + 90\right)\cdot 109 + \left(85 a + 21\right)\cdot 109^{2} + \left(71 a + 57\right)\cdot 109^{3} + \left(54 a + 44\right)\cdot 109^{4} + \left(21 a + 23\right)\cdot 109^{5} + \left(64 a + 18\right)\cdot 109^{6} + \left(33 a + 74\right)\cdot 109^{7} + \left(48 a + 60\right)\cdot 109^{8} + \left(6 a + 76\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 68 + 50\cdot 109 + 44\cdot 109^{2} + 86\cdot 109^{3} + 35\cdot 109^{4} + 3\cdot 109^{5} + 18\cdot 109^{6} + 87\cdot 109^{7} + 2\cdot 109^{8} + 27\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a + 106 + \left(93 a + 97\right)\cdot 109 + \left(74 a + 33\right)\cdot 109^{2} + \left(15 a + 30\right)\cdot 109^{3} + \left(75 a + 104\right)\cdot 109^{4} + \left(3 a + 30\right)\cdot 109^{5} + \left(93 a + 85\right)\cdot 109^{6} + \left(54 a + 86\right)\cdot 109^{7} + \left(71 a + 50\right)\cdot 109^{8} + \left(24 a + 47\right)\cdot 109^{9} +O(109^{10})\)
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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 |
| $1$ | $1$ | $()$ | $28$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-4$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $4$ |
| $112$ | $3$ | $(1,2,3)$ | $1$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $-2$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.