Properties

Label 3.257.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $257$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(257\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.257.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.257.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.0.257.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 73 + 47\cdot 157 + 11\cdot 157^{2} + 57\cdot 157^{3} + 146\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 75 + 34\cdot 157 + 88\cdot 157^{2} + 46\cdot 157^{3} + 3\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 77 + 138\cdot 157 + 50\cdot 157^{2} + 98\cdot 157^{3} + 122\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 89 + 93\cdot 157 + 6\cdot 157^{2} + 112\cdot 157^{3} + 41\cdot 157^{4} +O(157^{5})\) 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 valueComplex conjugation
$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$