Basic invariants
| Dimension: | $64$ |
| Group: | $A_8$ |
| Conductor: | \(916\!\cdots\!024\)\(\medspace = 2^{176} \cdot 23^{54} \cdot 43^{54} \cdot 137^{54} \cdot 389^{54} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.5620030628480917722529796097626934820084447048892416.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 168 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.5620030628480917722529796097626934820084447048892416.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825316 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 349 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 349 }$:
\( x^{2} + 348x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 109\cdot 349 + 166\cdot 349^{2} + 349^{3} + 28\cdot 349^{4} + 30\cdot 349^{5} + 328\cdot 349^{6} + 85\cdot 349^{7} + 132\cdot 349^{8} + 205\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 214 a + 175 + \left(44 a + 98\right)\cdot 349 + \left(345 a + 74\right)\cdot 349^{2} + \left(65 a + 112\right)\cdot 349^{3} + \left(96 a + 233\right)\cdot 349^{4} + \left(185 a + 26\right)\cdot 349^{5} + \left(283 a + 37\right)\cdot 349^{6} + \left(206 a + 292\right)\cdot 349^{7} + \left(96 a + 104\right)\cdot 349^{8} + \left(231 a + 139\right)\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 146 + 291\cdot 349 + 134\cdot 349^{2} + 186\cdot 349^{3} + 287\cdot 349^{5} + 120\cdot 349^{6} + 239\cdot 349^{7} + 217\cdot 349^{8} + 191\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 105\cdot 349 + 92\cdot 349^{2} + 197\cdot 349^{3} + 158\cdot 349^{4} + 327\cdot 349^{5} + 14\cdot 349^{6} + 40\cdot 349^{7} + 14\cdot 349^{8} + 170\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 309 a + 80 + \left(208 a + 170\right)\cdot 349 + \left(142 a + 6\right)\cdot 349^{2} + \left(239 a + 149\right)\cdot 349^{3} + \left(184 a + 176\right)\cdot 349^{4} + \left(127 a + 71\right)\cdot 349^{5} + \left(305 a + 309\right)\cdot 349^{6} + \left(250 a + 197\right)\cdot 349^{7} + \left(28 a + 287\right)\cdot 349^{8} + \left(a + 159\right)\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a + 40 + \left(140 a + 70\right)\cdot 349 + \left(206 a + 289\right)\cdot 349^{2} + \left(109 a + 245\right)\cdot 349^{3} + \left(164 a + 121\right)\cdot 349^{4} + \left(221 a + 14\right)\cdot 349^{5} + \left(43 a + 138\right)\cdot 349^{6} + \left(98 a + 143\right)\cdot 349^{7} + \left(320 a + 65\right)\cdot 349^{8} + \left(347 a + 132\right)\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 190 + 273\cdot 349 + 257\cdot 349^{2} + 321\cdot 349^{3} + 64\cdot 349^{4} + 174\cdot 349^{5} + 312\cdot 349^{6} + 181\cdot 349^{7} + 230\cdot 349^{8} + 123\cdot 349^{9} +O(349^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 135 a + 40 + \left(304 a + 278\right)\cdot 349 + \left(3 a + 25\right)\cdot 349^{2} + \left(283 a + 182\right)\cdot 349^{3} + \left(252 a + 263\right)\cdot 349^{4} + \left(163 a + 115\right)\cdot 349^{5} + \left(65 a + 135\right)\cdot 349^{6} + \left(142 a + 215\right)\cdot 349^{7} + \left(252 a + 343\right)\cdot 349^{8} + \left(117 a + 273\right)\cdot 349^{9} +O(349^{10})\)
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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 |
| $1$ | $1$ | $()$ | $64$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $1$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $-1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.