Basic invariants
| Dimension: | $4$ |
| Group: | $Z_8 : Z_8^\times$ |
| Conductor: | \(731025\)\(\medspace = 3^{4} \cdot 5^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.3125131875.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Z_8 : Z_8^\times$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2\times D_4$ |
| Projective stem field: | Galois closure of 8.0.18275625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} - 5x^{5} + 10x^{4} + 11x^{3} + 20x^{2} + 11x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 739 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 63 + 576\cdot 739 + 13\cdot 739^{2} + 311\cdot 739^{3} + 99\cdot 739^{4} + 589\cdot 739^{5} + 86\cdot 739^{6} + 9\cdot 739^{7} + 227\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 292 + 42\cdot 739 + 695\cdot 739^{2} + 582\cdot 739^{3} + 137\cdot 739^{4} + 322\cdot 739^{5} + 411\cdot 739^{6} + 263\cdot 739^{7} + 409\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 311 + 326\cdot 739 + 716\cdot 739^{2} + 62\cdot 739^{3} + 738\cdot 739^{4} + 385\cdot 739^{5} + 724\cdot 739^{6} + 687\cdot 739^{7} + 623\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 547 + 546\cdot 739 + 710\cdot 739^{2} + 56\cdot 739^{3} + 534\cdot 739^{4} + 487\cdot 739^{5} + 395\cdot 739^{6} + 204\cdot 739^{7} + 139\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 558 + 28\cdot 739 + 37\cdot 739^{2} + 308\cdot 739^{3} + 106\cdot 739^{4} + 15\cdot 739^{5} + 271\cdot 739^{6} + 576\cdot 739^{7} + 487\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 566 + 91\cdot 739 + 732\cdot 739^{2} + 275\cdot 739^{3} + 395\cdot 739^{4} + 551\cdot 739^{5} + 708\cdot 739^{6} + 628\cdot 739^{7} + 353\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 632 + 387\cdot 739 + 184\cdot 739^{2} + 96\cdot 739^{3} + 419\cdot 739^{4} + 161\cdot 739^{5} + 172\cdot 739^{6} + 14\cdot 739^{7} + 609\cdot 739^{8} +O(739^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 728 + 216\cdot 739 + 605\cdot 739^{2} + 522\cdot 739^{3} + 525\cdot 739^{4} + 442\cdot 739^{5} + 185\cdot 739^{6} + 571\cdot 739^{7} + 105\cdot 739^{8} +O(739^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(3,4)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,4)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,7)(2,4,6,3)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,4,6,3)$ | $0$ |
| $4$ | $4$ | $(1,6,5,2)(3,8,4,7)$ | $0$ |
| $4$ | $8$ | $(1,3,8,2,5,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,7,6,5,4,8,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.