Properties

Label 2.3332.4t3.b
Dimension $2$
Group $D_{4}$
Conductor $3332$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:\(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.13328.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ \( 3 + 9\cdot 13 + 12\cdot 13^{2} + 3\cdot 13^{3} + 5\cdot 13^{4} + 8\cdot 13^{5} + 13^{6} +O(13^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 11\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 2\cdot 13^{4} + 12\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 + 10\cdot 13 + 3\cdot 13^{2} + 9\cdot 13^{3} + 4\cdot 13^{4} + 8\cdot 13^{5} + 4\cdot 13^{6} +O(13^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 7\cdot 13 + 10\cdot 13^{2} + 3\cdot 13^{3} + 10\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})\) Copy content Toggle raw display

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

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

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.