Properties

Label 2.399.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $399$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.1197.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.399.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 5x^{2} + 3x + 9 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 7 + 84\cdot 97 + 69\cdot 97^{2} + 23\cdot 97^{3} + 14\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 72\cdot 97 + 71\cdot 97^{2} + 87\cdot 97^{3} + 46\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 74\cdot 97 + 18\cdot 97^{2} + 58\cdot 97^{3} + 81\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 55 + 60\cdot 97 + 33\cdot 97^{2} + 24\cdot 97^{3} + 51\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$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.