Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(3920\)\(\medspace = 2^{4} \cdot 5 \cdot 7^{2} \) |
Artin number field: | Galois closure of 8.0.12047257600.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{35})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 70\cdot 109 + 98\cdot 109^{2} + 57\cdot 109^{3} + 46\cdot 109^{4} +O(109^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 + 104\cdot 109^{2} + 41\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})\) |
$r_{ 3 }$ | $=$ | \( 28 + 56\cdot 109 + 99\cdot 109^{2} + 35\cdot 109^{3} + 41\cdot 109^{4} +O(109^{5})\) |
$r_{ 4 }$ | $=$ | \( 33 + 77\cdot 109 + 75\cdot 109^{2} + 31\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\) |
$r_{ 5 }$ | $=$ | \( 44 + 84\cdot 109 + 47\cdot 109^{2} + 108\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\) |
$r_{ 6 }$ | $=$ | \( 53 + 45\cdot 109 + 19\cdot 109^{2} + 32\cdot 109^{3} + 70\cdot 109^{4} +O(109^{5})\) |
$r_{ 7 }$ | $=$ | \( 75 + 91\cdot 109 + 24\cdot 109^{2} + 82\cdot 109^{3} + 80\cdot 109^{4} +O(109^{5})\) |
$r_{ 8 }$ | $=$ | \( 90 + 10\cdot 109 + 75\cdot 109^{2} + 45\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $-2$ | $-2$ |
$2$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ | $0$ |
$2$ | $2$ | $(1,7)(4,5)$ | $0$ | $0$ |
$2$ | $2$ | $(1,3)(2,7)(4,8)(5,6)$ | $0$ | $0$ |
$1$ | $4$ | $(1,4,7,5)(2,6,3,8)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,7,4)(2,8,3,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,2,7,3)(4,6,5,8)$ | $0$ | $0$ |
$2$ | $4$ | $(1,8,7,6)(2,5,3,4)$ | $0$ | $0$ |
$2$ | $4$ | $(1,5,7,4)(2,6,3,8)$ | $0$ | $0$ |