Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(297\!\cdots\!512\)\(\medspace = 2^{176} \cdot 102953^{39} \) |
| Artin stem field: | Galois closure of 8.0.319649416647163494229316963315979649024.1 |
| 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 8.0.319649416647163494229316963315979649024.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823617 \)
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The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{2} + 193x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 132 + 188\cdot 199 + 145\cdot 199^{2} + 128\cdot 199^{3} + 37\cdot 199^{4} + 151\cdot 199^{5} + 97\cdot 199^{6} + 39\cdot 199^{7} + 4\cdot 199^{8} + 192\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 63 a + 49 + \left(84 a + 99\right)\cdot 199 + \left(39 a + 172\right)\cdot 199^{2} + \left(191 a + 121\right)\cdot 199^{3} + \left(76 a + 185\right)\cdot 199^{4} + \left(61 a + 21\right)\cdot 199^{5} + \left(16 a + 41\right)\cdot 199^{6} + \left(98 a + 30\right)\cdot 199^{7} + \left(33 a + 31\right)\cdot 199^{8} + \left(120 a + 3\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 115 + 9\cdot 199 + 157\cdot 199^{2} + 75\cdot 199^{3} + 77\cdot 199^{4} + 117\cdot 199^{5} + 48\cdot 199^{6} + 69\cdot 199^{7} + 37\cdot 199^{8} + 125\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 a + 176 + \left(154 a + 63\right)\cdot 199 + \left(22 a + 175\right)\cdot 199^{2} + \left(173 a + 120\right)\cdot 199^{3} + \left(49 a + 142\right)\cdot 199^{4} + \left(6 a + 31\right)\cdot 199^{5} + \left(97 a + 85\right)\cdot 199^{6} + \left(70 a + 41\right)\cdot 199^{7} + \left(82 a + 131\right)\cdot 199^{8} + \left(114 a + 149\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 128 + 111\cdot 199 + 69\cdot 199^{2} + 26\cdot 199^{3} + 6\cdot 199^{4} + 157\cdot 199^{5} + 173\cdot 199^{6} + 159\cdot 199^{7} + 143\cdot 199^{8} + 102\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 137 + 27\cdot 199 + 190\cdot 199^{2} + 144\cdot 199^{3} + 19\cdot 199^{4} + 183\cdot 199^{5} + 8\cdot 199^{6} + 83\cdot 199^{7} + 157\cdot 199^{8} + 171\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 136 a + 29 + \left(114 a + 144\right)\cdot 199 + \left(159 a + 125\right)\cdot 199^{2} + \left(7 a + 35\right)\cdot 199^{3} + \left(122 a + 58\right)\cdot 199^{4} + \left(137 a + 114\right)\cdot 199^{5} + \left(182 a + 77\right)\cdot 199^{6} + \left(100 a + 5\right)\cdot 199^{7} + \left(165 a + 134\right)\cdot 199^{8} + \left(78 a + 93\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 157 a + 30 + \left(44 a + 151\right)\cdot 199 + \left(176 a + 157\right)\cdot 199^{2} + \left(25 a + 141\right)\cdot 199^{3} + \left(149 a + 69\right)\cdot 199^{4} + \left(192 a + 19\right)\cdot 199^{5} + \left(101 a + 64\right)\cdot 199^{6} + \left(128 a + 168\right)\cdot 199^{7} + \left(116 a + 156\right)\cdot 199^{8} + \left(84 a + 156\right)\cdot 199^{9} +O(199^{10})\)
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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 |
| $1$ | $1$ | $()$ | $45$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-3$ |
| $112$ | $3$ | $(1,2,3)$ | $0$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.