Properties

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

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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 4.2.55296.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{6})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ \( 1 + 4\cdot 5 + 4\cdot 5^{2} + 5^{3} + 3\cdot 5^{4} + 5^{5} + 3\cdot 5^{6} +O(5^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 4\cdot 5 + 3\cdot 5^{3} + 2\cdot 5^{5} +O(5^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 + 4\cdot 5^{2} + 5^{3} + 4\cdot 5^{4} + 2\cdot 5^{5} + 4\cdot 5^{6} +O(5^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 + 3\cdot 5^{3} + 5^{4} + 3\cdot 5^{5} + 5^{6} +O(5^{7})\) 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.