Properties

Label 3.5031049.6t8.a.a
Dimension $3$
Group $S_4$
Conductor $5031049$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(5031049\)\(\medspace = 2243^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.2243.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.2243.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - x^{2} - 3x - 1 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 9 + 57\cdot 97 + 42\cdot 97^{2} + 54\cdot 97^{3} + 37\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 50 + 40\cdot 97 + 4\cdot 97^{2} + 76\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 66 + 75\cdot 97 + 32\cdot 97^{2} + 54\cdot 97^{3} + 41\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 70 + 20\cdot 97 + 17\cdot 97^{2} + 9\cdot 97^{3} + 50\cdot 97^{4} +O(97^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,4)$$-1$
$6$$2$$(1,2)$$-1$
$8$$3$$(1,2,3)$$0$
$6$$4$$(1,2,3,4)$$1$

The blue line marks the conjugacy class containing complex conjugation.