Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(111\)\(\medspace = 3 \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.4107.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.111.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 5x^{2} - 3 \) . |
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 17 + 22\cdot 127 + 7\cdot 127^{2} + 118\cdot 127^{3} + 91\cdot 127^{4} +O(127^{5})\) |
$r_{ 2 }$ | $=$ | \( 50 + 5\cdot 127 + 108\cdot 127^{2} + 114\cdot 127^{3} + 87\cdot 127^{4} +O(127^{5})\) |
$r_{ 3 }$ | $=$ | \( 77 + 121\cdot 127 + 18\cdot 127^{2} + 12\cdot 127^{3} + 39\cdot 127^{4} +O(127^{5})\) |
$r_{ 4 }$ | $=$ | \( 110 + 104\cdot 127 + 119\cdot 127^{2} + 8\cdot 127^{3} + 35\cdot 127^{4} +O(127^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.