Properties

Label 2.768.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $768$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(768\)\(\medspace = 2^{8} \cdot 3 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.6144.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{-3})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 7 + 60\cdot 67 + 41\cdot 67^{2} + 28\cdot 67^{3} + 47\cdot 67^{4} +O(67^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 50\cdot 67 + 29\cdot 67^{2} + 37\cdot 67^{3} + 28\cdot 67^{4} +O(67^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 42 + 16\cdot 67 + 37\cdot 67^{2} + 29\cdot 67^{3} + 38\cdot 67^{4} +O(67^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 60 + 6\cdot 67 + 25\cdot 67^{2} + 38\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})\)  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.