Properties

Label 2.56.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $56$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(56\)\(\medspace = 2^{3} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.392.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.56.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{2}, \sqrt{-7})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 2\cdot 23 + 17\cdot 23^{2} + 22\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 19\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 7\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 9\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 6\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 21 + 14\cdot 23 + 21\cdot 23^{2} + 7\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

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.