Properties

Label 2.712.4t3.b
Dimension $2$
Group $D_{4}$
Conductor $712$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(712\)\(\medspace = 2^{3} \cdot 89 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.5696.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{89})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 8 + 21\cdot 97 + 21\cdot 97^{2} + 7\cdot 97^{3} + 49\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 24\cdot 97 + 77\cdot 97^{2} + 68\cdot 97^{3} + 80\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 60 + 12\cdot 97 + 25\cdot 97^{2} + 68\cdot 97^{3} + 96\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 73 + 38\cdot 97 + 70\cdot 97^{2} + 49\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})\)  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.