Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.53067.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 5\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{3} + 13^{4} + 5\cdot 13^{5} + 7\cdot 13^{6} +O(13^{7})\) |
$r_{ 2 }$ | $=$ | \( 6 + 12\cdot 13 + 6\cdot 13^{2} + 11\cdot 13^{3} + 8\cdot 13^{4} + 5\cdot 13^{5} + 13^{6} +O(13^{7})\) |
$r_{ 3 }$ | $=$ | \( 7 + 6\cdot 13^{2} + 13^{3} + 4\cdot 13^{4} + 7\cdot 13^{5} + 11\cdot 13^{6} +O(13^{7})\) |
$r_{ 4 }$ | $=$ | \( 12 + 7\cdot 13 + 2\cdot 13^{2} + 9\cdot 13^{3} + 11\cdot 13^{4} + 7\cdot 13^{5} + 5\cdot 13^{6} +O(13^{7})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |