Properties

Label 2.328.4t3.c
Dimension $2$
Group $D_{4}$
Conductor $328$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(328\)\(\medspace = 2^{3} \cdot 41 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.4.2624.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{41})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 19 + 37\cdot 73 + 2\cdot 73^{2} + 52\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 23 + 73 + 68\cdot 73^{2} + 11\cdot 73^{3} + 8\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 47 + 44\cdot 73 + 24\cdot 73^{2} + 31\cdot 73^{3} + 40\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 59 + 62\cdot 73 + 50\cdot 73^{2} + 50\cdot 73^{3} + 61\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,2)(3,4)$ $-2$
$2$ $2$ $(1,3)(2,4)$ $0$
$2$ $2$ $(1,2)$ $0$
$2$ $4$ $(1,4,2,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.