Properties

Label 2.111.4t3.a
Dimension $2$
Group $D_{4}$
Conductor $111$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(111\)\(\medspace = 3 \cdot 37 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.333.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-3}, \sqrt{37})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 45 + 77\cdot 127 + 15\cdot 127^{2} + 79\cdot 127^{3} + 89\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 50 + 20\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 117\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 58 + 67\cdot 127 + 109\cdot 127^{2} + 95\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 102 + 88\cdot 127 + 101\cdot 127^{2} + 57\cdot 127^{3} + 8\cdot 127^{4} +O(127^{5})\)  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 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.