Basic invariants
Dimension: | $2$ |
Group: | $QD_{16}$ |
Conductor: | \(441\)\(\medspace = 3^{2} \cdot 7^{2} \) |
Artin number field: | Galois closure of 8.2.257298363.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $QD_{16}$ |
Parity: | odd |
Projective image: | $D_4$ |
Projective field: | Galois closure of 4.0.189.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 39\cdot 79 + 56\cdot 79^{2} + 40\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 + 22\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\) |
$r_{ 3 }$ | $=$ | \( 13 + 13\cdot 79 + 3\cdot 79^{2} + 65\cdot 79^{3} + 28\cdot 79^{4} +O(79^{5})\) |
$r_{ 4 }$ | $=$ | \( 14 + 25\cdot 79 + 49\cdot 79^{2} + 17\cdot 79^{3} +O(79^{5})\) |
$r_{ 5 }$ | $=$ | \( 39 + 53\cdot 79 + 46\cdot 79^{2} + 48\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\) |
$r_{ 6 }$ | $=$ | \( 49 + 71\cdot 79 + 70\cdot 79^{2} + 52\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\) |
$r_{ 7 }$ | $=$ | \( 51 + 11\cdot 79 + 21\cdot 79^{2} + 34\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\) |
$r_{ 8 }$ | $=$ | \( 56 + 8\cdot 79^{2} + 10\cdot 79^{3} + 72\cdot 79^{4} +O(79^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ | $-2$ |
$4$ | $2$ | $(3,8)(4,6)(5,7)$ | $0$ | $0$ |
$2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ | $0$ |
$4$ | $4$ | $(1,8,2,7)(3,6,5,4)$ | $0$ | $0$ |
$2$ | $8$ | $(1,3,4,8,2,5,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,5,4,7,2,3,6,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |