Properties

Label 2.3520.4t3.c
Dimension $2$
Group $D_{4}$
Conductor $3520$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.17600.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 9 + 27\cdot 31 + 20\cdot 31^{2} + 24\cdot 31^{3} + 17\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 + 12\cdot 31 + 10\cdot 31^{2} + 13\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 18\cdot 31 + 20\cdot 31^{2} + 17\cdot 31^{3} + 17\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 3\cdot 31 + 10\cdot 31^{2} + 6\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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.