Properties

Label 2.799.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $799$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(799\)\(\medspace = 17 \cdot 47 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.13583.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.799.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{17}, \sqrt{-47})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 2 x^{3} + 9 x^{2} - 8 x - 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 16 + 18\cdot 53 + 36\cdot 53^{2} + 43\cdot 53^{3} + 27\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 11\cdot 53 + 51\cdot 53^{2} + 3\cdot 53^{3} + 29\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 29 + 41\cdot 53 + 53^{2} + 49\cdot 53^{3} + 23\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 38 + 34\cdot 53 + 16\cdot 53^{2} + 9\cdot 53^{3} + 25\cdot 53^{4} +O(53^{5})\)  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.