Properties

Label 2.819.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $819$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(819\)\(\medspace = 3^{2} \cdot 7 \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.2457.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.91.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-3}, \sqrt{-91})\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} - 6 x^{2} - x + 19\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 19 + 13\cdot 19^{2} + 13\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 14\cdot 19 + 13\cdot 19^{2} + 5\cdot 19^{3} + 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 6\cdot 19 + 4\cdot 19^{2} + 18\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 15\cdot 19 + 6\cdot 19^{2} + 19^{4} +O(19^{5})\)  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.