Properties

Label 2.3332.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $3332$
Root number $1$
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 stem field: Galois closure of 4.0.13328.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.68.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 3x^{2} - 14x + 53 \) Copy content Toggle raw display .

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 value
$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.