Properties

Label 3.751.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $751$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(751\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.751.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.751.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.751.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 32 + 98\cdot 163 + 133\cdot 163^{2} + 94\cdot 163^{3} + 70\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 34 + 7\cdot 163 + 75\cdot 163^{2} + 5\cdot 163^{3} + 99\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 118 + 112\cdot 163 + 142\cdot 163^{2} + 115\cdot 163^{3} + 75\cdot 163^{4} +O(163^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 144 + 107\cdot 163 + 137\cdot 163^{2} + 109\cdot 163^{3} + 80\cdot 163^{4} +O(163^{5})\) 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$