Properties

Label 3.8281.4t4.b.a
Dimension $3$
Group $A_4$
Conductor $8281$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(8281\)\(\medspace = 7^{2} \cdot 13^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.8281.1
Galois orbit size: $1$
Smallest permutation container: $A_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_4$
Projective stem field: 4.0.8281.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 13 + 33\cdot 97 + 63\cdot 97^{2} + 80\cdot 97^{3} + 69\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 31 + 76\cdot 97 + 57\cdot 97^{2} + 68\cdot 97^{3} + 80\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 64 + 71\cdot 97 + 39\cdot 97^{2} + 9\cdot 97^{3} + 81\cdot 97^{4} +O(97^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 87 + 12\cdot 97 + 33\cdot 97^{2} + 35\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.