Properties

Label 2.3143.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $3143$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(3143\)\(\medspace = 7 \cdot 449 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.1411207.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.3143.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-7}, \sqrt{449})\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{2} - 112\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 40 + 71\cdot 127 + 67\cdot 127^{2} + 41\cdot 127^{3} + 99\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 59 + 57\cdot 127 + 60\cdot 127^{2} + 69\cdot 127^{3} + 8\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 68 + 69\cdot 127 + 66\cdot 127^{2} + 57\cdot 127^{3} + 118\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 87 + 55\cdot 127 + 59\cdot 127^{2} + 85\cdot 127^{3} + 27\cdot 127^{4} +O(127^{5})\)  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.