Properties

Label 2.1440.4t3.a
Dimension $2$
Group $D_{4}$
Conductor $1440$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.21600.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{6}, \sqrt{-10})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 3 + 9\cdot 19 + 14\cdot 19^{2} + 17\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 13\cdot 19 + 2\cdot 19^{2} + 9\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 5\cdot 19 + 16\cdot 19^{2} + 9\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 9\cdot 19 + 4\cdot 19^{2} + 19^{3} + 13\cdot 19^{4} +O(19^{5})\)  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.