Properties

Label 2.387.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $387$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(387\)\(\medspace = 3^{2} \cdot 43 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.1161.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.43.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-43})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 6x^{2} - x + 7 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 89\cdot 97 + 21\cdot 97^{2} + 82\cdot 97^{3} + 6\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 57 + 91\cdot 97 + 87\cdot 97^{2} + 65\cdot 97^{3} + 79\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 58 + 55\cdot 97 + 81\cdot 97^{2} + 62\cdot 97^{3} + 58\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 76 + 54\cdot 97 + 2\cdot 97^{2} + 80\cdot 97^{3} + 48\cdot 97^{4} +O(97^{5})\) 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.