Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(603\)\(\medspace = 3^{2} \cdot 67 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.1809.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-67})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 83\cdot 151 + 74\cdot 151^{2} + 37\cdot 151^{3} + 30\cdot 151^{4} +O(151^{5})\)
|
$r_{ 2 }$ | $=$ |
\( 14 + 58\cdot 151 + 94\cdot 151^{2} + 67\cdot 151^{3} + 145\cdot 151^{4} +O(151^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 19 + 144\cdot 151 + 11\cdot 151^{2} + 149\cdot 151^{3} + 91\cdot 151^{4} +O(151^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 112 + 16\cdot 151 + 121\cdot 151^{2} + 47\cdot 151^{3} + 34\cdot 151^{4} +O(151^{5})\)
|
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |