Properties

Label 2.1400.4t3.c
Dimension $2$
Group $D_{4}$
Conductor $1400$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.9800.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{2}, \sqrt{-7})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 14 + 11\cdot 113 + 96\cdot 113^{2} + 105\cdot 113^{3} + 87\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 29 + 38\cdot 113 + 96\cdot 113^{2} + 16\cdot 113^{3} + 44\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 75 + 52\cdot 113 + 102\cdot 113^{2} + 43\cdot 113^{3} + 99\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 109 + 10\cdot 113 + 44\cdot 113^{2} + 59\cdot 113^{3} + 107\cdot 113^{4} +O\left(113^{ 5 }\right)$

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

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

Character values on conjugacy classes

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