Properties

Label 2.68.4t3.a
Dimension $2$
Group $D_{4}$
Conductor $68$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(68\)\(\medspace = 2^{2} \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.272.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{17})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 4 + 37\cdot 53 + 35\cdot 53^{2} + 40\cdot 53^{3} + 4\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 2\cdot 53 + 44\cdot 53^{2} + 24\cdot 53^{3} + 44\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 25\cdot 53 + 7\cdot 53^{2} + 8\cdot 53^{3} + 51\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 21 + 41\cdot 53 + 18\cdot 53^{2} + 32\cdot 53^{3} + 5\cdot 53^{4} +O(53^{5})\)  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.