Basic invariants
Dimension: | $56$ |
Group: | $A_8$ |
Conductor: | \(514\!\cdots\!416\)\(\medspace = 2^{156} \cdot 67^{48} \cdot 193^{48} \cdot 1019^{48} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.87812768645884871208063446530730170282275531390976.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 105 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_8$ |
Projective stem field: | Galois closure of 8.0.87812768645884871208063446530730170282275531390976.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825232 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 953 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 953 }$: \( x^{2} + 947x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 946 a + 96 + \left(128 a + 863\right)\cdot 953 + \left(350 a + 871\right)\cdot 953^{2} + \left(735 a + 214\right)\cdot 953^{3} + \left(339 a + 719\right)\cdot 953^{4} + \left(445 a + 45\right)\cdot 953^{5} + \left(421 a + 936\right)\cdot 953^{6} + \left(597 a + 295\right)\cdot 953^{7} + \left(741 a + 877\right)\cdot 953^{8} + \left(457 a + 860\right)\cdot 953^{9} +O(953^{10})\)
$r_{ 2 }$ |
$=$ |
\( 485 a + 426 + \left(615 a + 358\right)\cdot 953 + \left(321 a + 642\right)\cdot 953^{2} + \left(139 a + 518\right)\cdot 953^{3} + \left(389 a + 470\right)\cdot 953^{4} + \left(780 a + 564\right)\cdot 953^{5} + \left(110 a + 934\right)\cdot 953^{6} + \left(364 a + 88\right)\cdot 953^{7} + \left(132 a + 421\right)\cdot 953^{8} + \left(32 a + 868\right)\cdot 953^{9} +O(953^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 283 + 15\cdot 953 + 434\cdot 953^{2} + 82\cdot 953^{3} + 185\cdot 953^{4} + 595\cdot 953^{5} + 415\cdot 953^{6} + 145\cdot 953^{7} + 466\cdot 953^{8} + 814\cdot 953^{9} +O(953^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a + 54 + \left(824 a + 691\right)\cdot 953 + \left(602 a + 937\right)\cdot 953^{2} + \left(217 a + 464\right)\cdot 953^{3} + \left(613 a + 116\right)\cdot 953^{4} + \left(507 a + 472\right)\cdot 953^{5} + \left(531 a + 160\right)\cdot 953^{6} + \left(355 a + 600\right)\cdot 953^{7} + \left(211 a + 917\right)\cdot 953^{8} + \left(495 a + 6\right)\cdot 953^{9} +O(953^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 76 + 31\cdot 953 + 811\cdot 953^{2} + 674\cdot 953^{3} + 497\cdot 953^{4} + 23\cdot 953^{5} + 76\cdot 953^{6} + 877\cdot 953^{7} + 189\cdot 953^{8} + 717\cdot 953^{9} +O(953^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 816 + 58\cdot 953 + 269\cdot 953^{2} + 651\cdot 953^{3} + 122\cdot 953^{4} + 56\cdot 953^{5} + 70\cdot 953^{6} + 220\cdot 953^{7} + 674\cdot 953^{8} + 368\cdot 953^{9} +O(953^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 468 a + 477 + \left(337 a + 707\right)\cdot 953 + \left(631 a + 50\right)\cdot 953^{2} + \left(813 a + 80\right)\cdot 953^{3} + \left(563 a + 760\right)\cdot 953^{4} + \left(172 a + 92\right)\cdot 953^{5} + \left(842 a + 819\right)\cdot 953^{6} + \left(588 a + 256\right)\cdot 953^{7} + \left(820 a + 851\right)\cdot 953^{8} + \left(920 a + 928\right)\cdot 953^{9} +O(953^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 631 + 133\cdot 953 + 748\cdot 953^{2} + 171\cdot 953^{3} + 940\cdot 953^{4} + 55\cdot 953^{5} + 400\cdot 953^{6} + 374\cdot 953^{7} + 367\cdot 953^{8} + 199\cdot 953^{9} +O(953^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $56$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
$210$ | $2$ | $(1,2)(3,4)$ | $0$ |
$112$ | $3$ | $(1,2,3)$ | $-4$ |
$1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
$2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
$3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
$2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
$1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
$1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.