Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(17545\)\(\medspace = 5 \cdot 11^{2} \cdot 29 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.508805.3 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.145.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{29})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 16x^{2} + 32x + 67 \) . |
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 68 + 97\cdot 151 + 111\cdot 151^{2} + 81\cdot 151^{3} + 36\cdot 151^{4} +O(151^{5})\)
$r_{ 2 }$ |
$=$ |
\( 95 + 53\cdot 151 + 144\cdot 151^{2} + 29\cdot 151^{3} + 68\cdot 151^{4} +O(151^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 145 + 124\cdot 151 + 87\cdot 151^{3} + 124\cdot 151^{4} +O(151^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 146 + 25\cdot 151 + 45\cdot 151^{2} + 103\cdot 151^{3} + 72\cdot 151^{4} +O(151^{5})\)
| |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,3)$ | $0$ |
$2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.