Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(1575\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.1093955625.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 5x^{6} - 18x^{4} + 100x^{2} + 1 \) . |
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 20\cdot 37 + 32\cdot 37^{2} + 29\cdot 37^{3} + 2\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 10 + 13\cdot 37 + 30\cdot 37^{2} + 5\cdot 37^{3} + 10\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 12 + 8\cdot 37 + 29\cdot 37^{2} + 24\cdot 37^{3} + 18\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( 13 + 27\cdot 37 + 21\cdot 37^{2} + 4\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 + 9\cdot 37 + 15\cdot 37^{2} + 32\cdot 37^{3} + 9\cdot 37^{4} +O(37^{5})\) |
$r_{ 6 }$ | $=$ | \( 25 + 28\cdot 37 + 7\cdot 37^{2} + 12\cdot 37^{3} + 18\cdot 37^{4} +O(37^{5})\) |
$r_{ 7 }$ | $=$ | \( 27 + 23\cdot 37 + 6\cdot 37^{2} + 31\cdot 37^{3} + 26\cdot 37^{4} +O(37^{5})\) |
$r_{ 8 }$ | $=$ | \( 31 + 16\cdot 37 + 4\cdot 37^{2} + 7\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
$2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
$2$ | $4$ | $(1,2,4,6)(3,8,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.