Properties

Label 2.663.4t3.a
Dimension $2$
Group $D_{4}$
Conductor $663$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(663\)\(\medspace = 3 \cdot 13 \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.8619.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{13}, \sqrt{-51})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 2 + 11\cdot 23 + 7\cdot 23^{2} + 18\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 14\cdot 23 + 6\cdot 23^{2} + 14\cdot 23^{3} + 7\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 23 + 7\cdot 23^{2} + 9\cdot 23^{3} + 16\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 19\cdot 23 + 23^{2} + 22\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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