Properties

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

Related objects

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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 number field: Galois closure of 4.0.3072.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)

Galois action

Roots of defining polynomial

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\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 25 + 25\cdot 59 + 44\cdot 59^{2} + 17\cdot 59^{3} + 12\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 34 + 33\cdot 59 + 14\cdot 59^{2} + 41\cdot 59^{3} + 46\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 50 + 56\cdot 59 + 23\cdot 59^{2} + 32\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$

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.