Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(111\)\(\medspace = 3 \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.333.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
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 $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 45 + 77\cdot 127 + 15\cdot 127^{2} + 79\cdot 127^{3} + 89\cdot 127^{4} +O(127^{5})\) |
$r_{ 2 }$ | $=$ | \( 50 + 20\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 117\cdot 127^{4} +O(127^{5})\) |
$r_{ 3 }$ | $=$ | \( 58 + 67\cdot 127 + 109\cdot 127^{2} + 95\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\) |
$r_{ 4 }$ | $=$ | \( 102 + 88\cdot 127 + 101\cdot 127^{2} + 57\cdot 127^{3} + 8\cdot 127^{4} +O(127^{5})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |