Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.796594176.12 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.56.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 5x^{6} - 4x^{5} + x^{4} - 18x^{3} + 15x^{2} + 18 \) . |
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 28\cdot 71 + 60\cdot 71^{2} + 46\cdot 71^{3} + 20\cdot 71^{4} +O(71^{5})\) |
$r_{ 2 }$ | $=$ | \( 5 + 14\cdot 71 + 59\cdot 71^{2} + 2\cdot 71^{3} + 30\cdot 71^{4} +O(71^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 + 11\cdot 71 + 4\cdot 71^{2} + 20\cdot 71^{3} + 32\cdot 71^{4} +O(71^{5})\) |
$r_{ 4 }$ | $=$ | \( 10 + 59\cdot 71 + 69\cdot 71^{2} + 61\cdot 71^{3} + 67\cdot 71^{4} +O(71^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 + 67\cdot 71 + 48\cdot 71^{2} + 18\cdot 71^{3} + 67\cdot 71^{4} +O(71^{5})\) |
$r_{ 6 }$ | $=$ | \( 27 + 41\cdot 71 + 59\cdot 71^{2} + 6\cdot 71^{3} + 34\cdot 71^{4} +O(71^{5})\) |
$r_{ 7 }$ | $=$ | \( 31 + 4\cdot 71 + 19\cdot 71^{2} + 41\cdot 71^{3} + 45\cdot 71^{4} +O(71^{5})\) |
$r_{ 8 }$ | $=$ | \( 38 + 58\cdot 71 + 33\cdot 71^{2} + 14\cdot 71^{3} + 57\cdot 71^{4} +O(71^{5})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-2$ |
$2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
$2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
$2$ | $4$ | $(1,2,3,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.