Properties

Label 2.95.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $95$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(95\)\(\medspace = 5 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.475.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.95.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{5}, \sqrt{-19})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 35 + 56\cdot 131 + 56\cdot 131^{2} + 104\cdot 131^{3} +O(131^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 40 + 47\cdot 131 + 85\cdot 131^{2} + 72\cdot 131^{3} + 46\cdot 131^{4} +O(131^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 92 + 83\cdot 131 + 45\cdot 131^{2} + 58\cdot 131^{3} + 84\cdot 131^{4} +O(131^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 97 + 74\cdot 131 + 74\cdot 131^{2} + 26\cdot 131^{3} + 130\cdot 131^{4} +O(131^{5})\)  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.