Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(128\)\(\medspace = 2^{7} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.0.4194304.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 11 + 19\cdot 41 + 9\cdot 41^{2} + 18\cdot 41^{3} + 15\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 + 32\cdot 41 + 15\cdot 41^{2} + 29\cdot 41^{3} + 14\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 15 + 11\cdot 41 + 20\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 17 + 26\cdot 41 + 17\cdot 41^{2} + 39\cdot 41^{3} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 + 14\cdot 41 + 23\cdot 41^{2} + 41^{3} + 40\cdot 41^{4} +O(41^{5})\) |
$r_{ 6 }$ | $=$ | \( 26 + 29\cdot 41 + 20\cdot 41^{2} + 5\cdot 41^{3} + 5\cdot 41^{4} +O(41^{5})\) |
$r_{ 7 }$ | $=$ | \( 29 + 8\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 26\cdot 41^{4} +O(41^{5})\) |
$r_{ 8 }$ | $=$ | \( 30 + 21\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
$2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |