Properties

Label 2.3743.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $3743$
Root number $1$
Indicator $1$

Related objects

Learn more about

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(3743\)\(\medspace = 19 \cdot 197 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.71117.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.3743.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-19}, \sqrt{197})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 11\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 8 + 45\cdot 47 + 27\cdot 47^{2} + 9\cdot 47^{3} + 43\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 41\cdot 47 + 17\cdot 47^{2} + 34\cdot 47^{3} + 32\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 38 + 5\cdot 47 + 29\cdot 47^{2} + 12\cdot 47^{3} + 14\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 40 + 47 + 19\cdot 47^{2} + 37\cdot 47^{3} + 3\cdot 47^{4} +O(47^{5})\)  Toggle raw display

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

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

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.