Properties

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

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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 stem field: 4.0.272.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.68.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{17})\)

Defining polynomial

$f(x)$$=$\(x^{4} + x^{2} - 2 x + 1\)  Toggle raw display.

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 value
$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.