Basic invariants
Dimension: | $2$ |
Group: | $Q_8:C_2$ |
Conductor: | \(2775\)\(\medspace = 3 \cdot 5^{2} \cdot 37 \) |
Artin number field: | Galois closure of 8.0.1732640625.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $Q_8:C_2$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 114\cdot 139 + 66\cdot 139^{2} + 9\cdot 139^{3} + 19\cdot 139^{4} +O(139^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 + 2\cdot 139 + 82\cdot 139^{2} + 134\cdot 139^{3} + 61\cdot 139^{4} +O(139^{5})\) |
$r_{ 3 }$ | $=$ | \( 25 + 88\cdot 139 + 116\cdot 139^{2} + 29\cdot 139^{3} + 81\cdot 139^{4} +O(139^{5})\) |
$r_{ 4 }$ | $=$ | \( 39 + 65\cdot 139 + 51\cdot 139^{2} + 40\cdot 139^{3} + 54\cdot 139^{4} +O(139^{5})\) |
$r_{ 5 }$ | $=$ | \( 61 + 39\cdot 139 + 101\cdot 139^{2} + 60\cdot 139^{3} + 116\cdot 139^{4} +O(139^{5})\) |
$r_{ 6 }$ | $=$ | \( 69 + 122\cdot 139 + 27\cdot 139^{2} + 73\cdot 139^{3} + 80\cdot 139^{4} +O(139^{5})\) |
$r_{ 7 }$ | $=$ | \( 87 + 27\cdot 139 + 17\cdot 139^{2} + 84\cdot 139^{3} + 137\cdot 139^{4} +O(139^{5})\) |
$r_{ 8 }$ | $=$ | \( 128 + 96\cdot 139 + 92\cdot 139^{2} + 123\cdot 139^{3} + 4\cdot 139^{4} +O(139^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-2$ | $-2$ |
$2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ | $0$ |
$2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ | $0$ |
$2$ | $2$ | $(2,6)(3,4)$ | $0$ | $0$ |
$1$ | $4$ | $(1,7,5,8)(2,3,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,8,5,7)(2,4,6,3)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,3,5,4)(2,7,6,8)$ | $0$ | $0$ |
$2$ | $4$ | $(1,7,5,8)(2,4,6,3)$ | $0$ | $0$ |
$2$ | $4$ | $(1,2,5,6)(3,8,4,7)$ | $0$ | $0$ |