Basic invariants
| Dimension: | $4$ |
| Group: | $(C_8:C_2):C_2$ |
| Conductor: | \(600625\)\(\medspace = 5^{4} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.75078125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $(C_8:C_2):C_2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2:C_4$ |
| Projective stem field: | Galois closure of 8.0.15015625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 7x^{6} + 11x^{5} + 20x^{4} - 14x^{3} - 22x^{2} + 3x + 11 \)
|
The roots of $f$ are computed in $\Q_{ 691 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 387 + 637\cdot 691 + 635\cdot 691^{2} + 584\cdot 691^{3} +O(691^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 428 + 44\cdot 691 + 93\cdot 691^{2} + 400\cdot 691^{3} + 608\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 481 + 117\cdot 691 + 457\cdot 691^{2} + 249\cdot 691^{3} + 460\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 485 + 298\cdot 691 + 73\cdot 691^{2} + 364\cdot 691^{3} + 212\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 526 + 396\cdot 691 + 221\cdot 691^{2} + 179\cdot 691^{3} + 129\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 577 + 607\cdot 691 + 667\cdot 691^{2} + 322\cdot 691^{3} + 327\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 584 + 430\cdot 691 + 547\cdot 691^{2} + 294\cdot 691^{3} + 233\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 680 + 229\cdot 691 + 67\cdot 691^{2} + 368\cdot 691^{3} + 100\cdot 691^{4} +O(691^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,4)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,5,4)(3,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,2,5,4)(3,6,8,7)$ | $0$ |
| $4$ | $8$ | $(1,6,2,3,5,7,4,8)$ | $0$ |
| $4$ | $8$ | $(1,3,4,6,5,8,2,7)$ | $0$ |
| $4$ | $8$ | $(1,3,2,6,5,8,4,7)$ | $0$ |
| $4$ | $8$ | $(1,6,4,3,5,7,2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.