Properties

Label 2.7_11_43.4t3.4
Dimension 2
Group $D_{4}$
Conductor $ 7 \cdot 11 \cdot 43 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$3311= 7 \cdot 11 \cdot 43 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 32 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 30\cdot 53 + 11\cdot 53^{2} + 20\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 39\cdot 53 + 3\cdot 53^{2} + 34\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 35 + 13\cdot 53 + 49\cdot 53^{2} + 18\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 + 22\cdot 53 + 41\cdot 53^{2} + 32\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,4)(2,3)$ $-2$
$2$ $2$ $(1,2)(3,4)$ $0$
$2$ $2$ $(1,4)$ $0$
$2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.