Properties

Label 2.384.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $384$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(384\)\(\medspace = 2^{7} \cdot 3 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.3072.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.24.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{3})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 2\cdot 59 + 35\cdot 59^{2} + 26\cdot 59^{3} + 23\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 25\cdot 59 + 44\cdot 59^{2} + 17\cdot 59^{3} + 12\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 34 + 33\cdot 59 + 14\cdot 59^{2} + 41\cdot 59^{3} + 46\cdot 59^{4} +O(59^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 56\cdot 59 + 23\cdot 59^{2} + 32\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\)  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.