Properties

Label 2.2304.4t3.e.a
Dimension $2$
Group $D_{4}$
Conductor $2304$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.55296.4
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{6})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 24 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 19\cdot 29 + 28\cdot 29^{2} + 14\cdot 29^{3} + 27\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 27\cdot 29 + 23\cdot 29^{2} + 21\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 29 + 5\cdot 29^{2} + 7\cdot 29^{3} + 26\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 25 + 9\cdot 29 + 14\cdot 29^{3} + 29^{4} +O(29^{5})\) 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.