Basic invariants
| Dimension: | $35$ |
| Group: | $A_8$ |
| Conductor: | \(319\!\cdots\!016\)\(\medspace = 2^{134} \cdot 7^{42} \cdot 11^{30} \cdot 191^{30} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.133100753213221593424899389161209856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.133100753213221593424899389161209856.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823585 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 121 a + 67 + \left(178 a + 135\right)\cdot 193 + \left(181 a + 110\right)\cdot 193^{2} + \left(150 a + 160\right)\cdot 193^{3} + \left(66 a + 82\right)\cdot 193^{4} + \left(138 a + 189\right)\cdot 193^{5} + \left(151 a + 164\right)\cdot 193^{6} + \left(46 a + 192\right)\cdot 193^{7} + \left(114 a + 92\right)\cdot 193^{8} + \left(140 a + 174\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 76 + 117\cdot 193 + 102\cdot 193^{2} + 88\cdot 193^{3} + 74\cdot 193^{4} + 103\cdot 193^{5} + 84\cdot 193^{6} + 153\cdot 193^{7} + 141\cdot 193^{8} + 91\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 191 + \left(162 a + 14\right)\cdot 193 + \left(68 a + 131\right)\cdot 193^{2} + \left(114 a + 104\right)\cdot 193^{3} + \left(154 a + 71\right)\cdot 193^{4} + \left(133 a + 166\right)\cdot 193^{5} + \left(132 a + 13\right)\cdot 193^{6} + \left(173 a + 85\right)\cdot 193^{7} + \left(21 a + 87\right)\cdot 193^{8} + \left(98 a + 104\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 114 + 166\cdot 193 + 29\cdot 193^{2} + 89\cdot 193^{3} + 134\cdot 193^{4} + 74\cdot 193^{5} + 150\cdot 193^{6} + 70\cdot 193^{7} + 65\cdot 193^{8} + 5\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 182 a + 9 + \left(30 a + 166\right)\cdot 193 + \left(124 a + 37\right)\cdot 193^{2} + \left(78 a + 150\right)\cdot 193^{3} + \left(38 a + 111\right)\cdot 193^{4} + \left(59 a + 145\right)\cdot 193^{5} + \left(60 a + 12\right)\cdot 193^{6} + \left(19 a + 126\right)\cdot 193^{7} + \left(171 a + 128\right)\cdot 193^{8} + \left(94 a + 180\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 95 + 15\cdot 193 + 126\cdot 193^{2} + 150\cdot 193^{3} + 70\cdot 193^{4} + 162\cdot 193^{5} + 41\cdot 193^{6} + 48\cdot 193^{7} + 121\cdot 193^{8} + 17\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 72 a + 188 + \left(14 a + 192\right)\cdot 193 + \left(11 a + 113\right)\cdot 193^{2} + \left(42 a + 129\right)\cdot 193^{3} + \left(126 a + 191\right)\cdot 193^{4} + \left(54 a + 67\right)\cdot 193^{5} + \left(41 a + 178\right)\cdot 193^{6} + \left(146 a + 87\right)\cdot 193^{7} + \left(78 a + 160\right)\cdot 193^{8} + \left(52 a + 7\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 + 156\cdot 193 + 119\cdot 193^{2} + 91\cdot 193^{3} + 34\cdot 193^{4} + 55\cdot 193^{5} + 125\cdot 193^{6} + 7\cdot 193^{7} + 167\cdot 193^{8} + 189\cdot 193^{9} +O(193^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $35$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-5$ |
| $112$ | $3$ | $(1,2,3)$ | $5$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $-1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.