Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(163\!\cdots\!696\)\(\medspace = 2^{138} \cdot 67^{39} \cdot 193^{39} \cdot 1019^{39} \) |
| Artin stem field: | Galois closure of 8.0.87812768645884871208063446530730170282275531390976.1 |
| 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 8.0.87812768645884871208063446530730170282275531390976.1 |
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 |
| $1$ | $1$ | $()$ | $45$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-3$ |
| $112$ | $3$ | $(1,2,3)$ | $0$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.