Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.1040.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{65})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 9 + 11\cdot 97 + 23\cdot 97^{2} + 83\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 + 44\cdot 97 + 62\cdot 97^{2} + 38\cdot 97^{3} + 33\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 29 + 23\cdot 97 + 65\cdot 97^{2} + 50\cdot 97^{3} + 3\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 46 + 18\cdot 97 + 43\cdot 97^{2} + 21\cdot 97^{3} +O(97^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)$ | $0$ |
$2$ | $2$ | $(1,2)$ | $0$ |
$2$ | $4$ | $(1,4,2,3)$ | $0$ |