Properties

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

Related objects

Learn more

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 stem field: 4.0.21600.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.40.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{6}, \sqrt{-10})\)

Defining polynomial

$f(x)$$=$\(x^{4} + 3 x^{2} + 6\)  Toggle raw display.

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 value
$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.