Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(295\)\(\medspace = 5 \cdot 59 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.1475.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-59})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 52\cdot 79 + 39\cdot 79^{2} + 67\cdot 79^{3} + 43\cdot 79^{4} +O(79^{5})\) |
$r_{ 2 }$ | $=$ | \( 33 + 51\cdot 79 + 68\cdot 79^{2} + 46\cdot 79^{3} + 78\cdot 79^{4} +O(79^{5})\) |
$r_{ 3 }$ | $=$ | \( 42 + 61\cdot 79 + 62\cdot 79^{2} + 57\cdot 79^{3} + 26\cdot 79^{4} +O(79^{5})\) |
$r_{ 4 }$ | $=$ | \( 76 + 71\cdot 79 + 65\cdot 79^{2} + 64\cdot 79^{3} + 8\cdot 79^{4} +O(79^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,3)$ | $0$ |
$2$ | $4$ | $(1,4,3,2)$ | $0$ |