Properties

Label 2.544.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $544$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(544\)\(\medspace = 2^{5} \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.4.4352.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.136.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{17})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 15 + 66\cdot 89 + 38\cdot 89^{2} + 73\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 29 + 2\cdot 89 + 72\cdot 89^{2} + 61\cdot 89^{3} + 8\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 88\cdot 89 + 10\cdot 89^{2} + 39\cdot 89^{3} + 79\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 85 + 20\cdot 89 + 56\cdot 89^{2} + 76\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display

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

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

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.