Properties

Label 2.552.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $552$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(552\)\(\medspace = 2^{3} \cdot 3 \cdot 23 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.12696.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.552.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{6}, \sqrt{-23})\)

Defining polynomial

$f(x)$$=$\(x^{4} + x^{2} + 6\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 7\cdot 71 + 23\cdot 71^{2} + 6\cdot 71^{3} + 47\cdot 71^{4} +O(71^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 34 + 10\cdot 71 + 33\cdot 71^{2} + 30\cdot 71^{3} + 62\cdot 71^{4} +O(71^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 37 + 60\cdot 71 + 37\cdot 71^{2} + 40\cdot 71^{3} + 8\cdot 71^{4} +O(71^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 60 + 63\cdot 71 + 47\cdot 71^{2} + 64\cdot 71^{3} + 23\cdot 71^{4} +O(71^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$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.