Properties

Label 2.2304.4t3.a
Dimension $2$
Group $D_{4}$
Conductor $2304$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.18432.3
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{8})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 1 + 13\cdot 17 + 5\cdot 17^{2} + 11\cdot 17^{3} + 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 3\cdot 17 + 8\cdot 17^{2} + 5\cdot 17^{3} + 11\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 13\cdot 17 + 8\cdot 17^{2} + 11\cdot 17^{3} + 5\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 3\cdot 17 + 11\cdot 17^{2} + 5\cdot 17^{3} + 15\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,4)$
$(1,2)(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,4)(2,3)$ $-2$
$2$ $2$ $(1,2)(3,4)$ $0$
$2$ $2$ $(1,4)$ $0$
$2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.