Properties

Label 2.1764.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $1764$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.12348.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{7})\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} - 4 x^{2} + 10 x + 16\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 15 + 6\cdot 29 + 3\cdot 29^{2} + 14\cdot 29^{3} + 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 22 + 18\cdot 29 + 26\cdot 29^{2} + 28\cdot 29^{3} + 3\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 25 + 7\cdot 29 + 29^{2} + 22\cdot 29^{3} + 19\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 26 + 24\cdot 29 + 26\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)$$-2$
$2$$2$$(1,3)(2,4)$$0$
$2$$2$$(1,2)$$0$
$2$$4$$(1,4,2,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.