Properties

Label 2.4647.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $4647$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(4647\)\(\medspace = 3 \cdot 1549 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.7198203.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.4647.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{1549})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 31x^{2} - 147 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 4 + 28\cdot 43 + 31\cdot 43^{2} + 5\cdot 43^{3} + 11\cdot 43^{4} + 39\cdot 43^{5} +O(43^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 21\cdot 43 + 6\cdot 43^{2} + 25\cdot 43^{3} + 24\cdot 43^{4} + 11\cdot 43^{5} +O(43^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 31 + 21\cdot 43 + 36\cdot 43^{2} + 17\cdot 43^{3} + 18\cdot 43^{4} + 31\cdot 43^{5} +O(43^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 + 14\cdot 43 + 11\cdot 43^{2} + 37\cdot 43^{3} + 31\cdot 43^{4} + 3\cdot 43^{5} +O(43^{6})\) Copy content 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.