Properties

Label 3.5031049.6t8.a
Dimension $3$
Group $S_4$
Conductor $5031049$
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 number field: Galois closure of 4.2.2243.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Projective image: $S_4$
Projective field: Galois closure of 4.2.2243.1

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $3$
$3$ $2$ $(1,2)(3,4)$ $-1$
$6$ $2$ $(1,2)$ $-1$
$8$ $3$ $(1,2,3)$ $0$
$6$ $4$ $(1,2,3,4)$ $1$
The blue line marks the conjugacy class containing complex conjugation.