Basic invariants
| Dimension: | $70$ |
| Group: | $A_8$ |
| Conductor: | \(162\!\cdots\!736\)\(\medspace = 2^{216} \cdot 701^{60} \cdot 18797^{60} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.87813088530439533539051393535468530591915378212864.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 120 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.87813088530439533539051393535468530591915378212864.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825360 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 593 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 593 }$:
\( x^{2} + 592x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 359 a + 470 + \left(268 a + 205\right)\cdot 593 + \left(125 a + 336\right)\cdot 593^{2} + \left(389 a + 184\right)\cdot 593^{3} + \left(531 a + 277\right)\cdot 593^{4} + \left(336 a + 455\right)\cdot 593^{5} + \left(209 a + 360\right)\cdot 593^{6} + \left(357 a + 54\right)\cdot 593^{7} + \left(333 a + 173\right)\cdot 593^{8} + \left(105 a + 59\right)\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 234 a + 236 + \left(324 a + 115\right)\cdot 593 + \left(467 a + 193\right)\cdot 593^{2} + \left(203 a + 448\right)\cdot 593^{3} + \left(61 a + 419\right)\cdot 593^{4} + \left(256 a + 260\right)\cdot 593^{5} + \left(383 a + 233\right)\cdot 593^{6} + \left(235 a + 202\right)\cdot 593^{7} + \left(259 a + 149\right)\cdot 593^{8} + \left(487 a + 424\right)\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 501 + 87\cdot 593 + 279\cdot 593^{2} + 419\cdot 593^{3} + 526\cdot 593^{4} + 192\cdot 593^{5} + 575\cdot 593^{6} + 95\cdot 593^{7} + 382\cdot 593^{8} + 256\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 89 + 49\cdot 593 + 578\cdot 593^{2} + 3\cdot 593^{3} + 581\cdot 593^{4} + 27\cdot 593^{5} + 381\cdot 593^{6} + 535\cdot 593^{7} + 482\cdot 593^{8} + 190\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 255 + 135\cdot 593 + 337\cdot 593^{2} + 590\cdot 593^{3} + 156\cdot 593^{4} + 22\cdot 593^{5} + 592\cdot 593^{6} + 248\cdot 593^{7} + 514\cdot 593^{8} + 460\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 205 + \left(262 a + 252\right)\cdot 593 + \left(510 a + 33\right)\cdot 593^{2} + \left(183 a + 191\right)\cdot 593^{3} + \left(266 a + 47\right)\cdot 593^{4} + \left(413 a + 411\right)\cdot 593^{5} + \left(491 a + 338\right)\cdot 593^{6} + \left(227 a + 413\right)\cdot 593^{7} + \left(431 a + 319\right)\cdot 593^{8} + \left(477 a + 340\right)\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 381 + 448\cdot 593 + 332\cdot 593^{2} + 76\cdot 593^{3} + 233\cdot 593^{4} + 443\cdot 593^{5} + 66\cdot 593^{6} + 78\cdot 593^{7} + 420\cdot 593^{8} + 252\cdot 593^{9} +O(593^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 563 a + 235 + \left(330 a + 484\right)\cdot 593 + \left(82 a + 281\right)\cdot 593^{2} + \left(409 a + 457\right)\cdot 593^{3} + \left(326 a + 129\right)\cdot 593^{4} + \left(179 a + 558\right)\cdot 593^{5} + \left(101 a + 416\right)\cdot 593^{6} + \left(365 a + 149\right)\cdot 593^{7} + \left(161 a + 523\right)\cdot 593^{8} + \left(115 a + 386\right)\cdot 593^{9} +O(593^{10})\)
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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 |
| $1$ | $1$ | $()$ | $70$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $112$ | $3$ | $(1,2,3)$ | $-5$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $-2$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.