Properties

Label 2.63.4t3.a
Dimension 2
Group $D_{4}$
Conductor $ 3^{2} \cdot 7 $
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$63= 3^{2} \cdot 7 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 22\cdot 67 + 31\cdot 67^{2} + 41\cdot 67^{3} + 49\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 + 54\cdot 67 + 30\cdot 67^{2} + 61\cdot 67^{3} + 5\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 26 + 55\cdot 67 + 57\cdot 67^{2} + 24\cdot 67^{3} + 59\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 + 2\cdot 67 + 14\cdot 67^{2} + 6\cdot 67^{3} + 19\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

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