Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr C_2\wr C_2$ |
| Conductor: | \(964975\)\(\medspace = 5^{2} \cdot 11^{3} \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.2193125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
| Parity: | odd |
| Projective image: | $C_2\wr C_2^2$ |
| Projective field: | Galois closure of 8.0.307827025.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 87 + 443\cdot 929 + 885\cdot 929^{2} + 341\cdot 929^{3} + 239\cdot 929^{4} + 712\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 97 + 391\cdot 929 + 161\cdot 929^{2} + 803\cdot 929^{3} + 490\cdot 929^{4} + 815\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 233 + 383\cdot 929 + 561\cdot 929^{2} + 307\cdot 929^{3} + 682\cdot 929^{4} + 669\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 299 + 474\cdot 929 + 74\cdot 929^{2} + 212\cdot 929^{3} + 372\cdot 929^{4} + 757\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 311 + 557\cdot 929 + 336\cdot 929^{2} + 67\cdot 929^{3} + 564\cdot 929^{4} + 647\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 358 + 377\cdot 929 + 73\cdot 929^{2} + 545\cdot 929^{3} + 634\cdot 929^{4} + 524\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 680 + 821\cdot 929 + 251\cdot 929^{2} + 697\cdot 929^{3} + 420\cdot 929^{4} + 482\cdot 929^{5} +O(929^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 724 + 267\cdot 929 + 442\cdot 929^{2} + 741\cdot 929^{3} + 311\cdot 929^{4} + 35\cdot 929^{5} +O(929^{6})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-4$ |
| $2$ | $2$ | $(2,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(2,7)$ | $2$ |
| $4$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,5)$ | $0$ |
| $4$ | $2$ | $(2,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $2$ |
| $8$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $0$ |
| $8$ | $2$ | $(1,4)(2,6)(7,8)$ | $0$ |
| $4$ | $4$ | $(1,3,4,5)(2,8,7,6)$ | $0$ |
| $4$ | $4$ | $(2,8,7,6)$ | $-2$ |
| $4$ | $4$ | $(1,5,4,3)(2,7)(6,8)$ | $2$ |
| $8$ | $4$ | $(1,6,4,8)(2,5,7,3)$ | $0$ |
| $8$ | $4$ | $(1,4)(2,8,7,6)$ | $0$ |
| $8$ | $4$ | $(1,3)(2,8,7,6)(4,5)$ | $0$ |
| $16$ | $4$ | $(1,6,3,2)(4,8,5,7)$ | $0$ |
| $16$ | $4$ | $(1,6)(2,5,7,3)(4,8)$ | $0$ |
| $16$ | $8$ | $(1,6,3,2,4,8,5,7)$ | $0$ |