Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(363\)\(\medspace = 3 \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.3993.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 23\cdot 97 + 27\cdot 97^{2} + 2\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 33 + 55\cdot 97 + 10\cdot 97^{3} + 43\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 58 + 60\cdot 97 + 72\cdot 97^{2} + 45\cdot 97^{3} + 23\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 94 + 54\cdot 97 + 93\cdot 97^{2} + 38\cdot 97^{3} + 63\cdot 97^{4} +O(97^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,3)$ | $0$ |
$2$ | $4$ | $(1,4,3,2)$ | $0$ |