Properties

Label 2.392.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $392$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(392\)\(\medspace = 2^{3} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.2744.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-7})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 15 + 19\cdot 67 + 64\cdot 67^{2} + 60\cdot 67^{3} + 8\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 34 + 41\cdot 67 + 41\cdot 67^{2} + 4\cdot 67^{3} + 33\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 41 + 50\cdot 67 + 14\cdot 67^{2} + 41\cdot 67^{3} + 3\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 45 + 22\cdot 67 + 13\cdot 67^{2} + 27\cdot 67^{3} + 21\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,3)(2,4)$$-2$
$2$$2$$(1,2)(3,4)$$0$
$2$$2$$(1,3)$$0$
$2$$4$$(1,4,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.