Properties

Label 2.2248.4t3.d.a
Dimension $2$
Group $D_{4}$
Conductor $2248$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2248\)\(\medspace = 2^{3} \cdot 281 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.631688.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.2248.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{281})\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 9x^{2} - 50 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 25 + 80\cdot 283 + 155\cdot 283^{2} + 69\cdot 283^{3} + 145\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 82 + 255\cdot 283 + 46\cdot 283^{2} + 254\cdot 283^{3} + 259\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 201 + 27\cdot 283 + 236\cdot 283^{2} + 28\cdot 283^{3} + 23\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 258 + 202\cdot 283 + 127\cdot 283^{2} + 213\cdot 283^{3} + 137\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display

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

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

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.