Properties

Label 3.338724.6t8.b.a
Dimension $3$
Group $S_4$
Conductor $338724$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(338724\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 97^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.10476.1
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 4.2.10476.1

Defining polynomial

$f(x)$$=$ \( x^{4} - 6x - 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 20 + 8\cdot 149 + 147\cdot 149^{2} + 12\cdot 149^{3} + 19\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 48 + 87\cdot 149 + 100\cdot 149^{2} + 65\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 110 + 84\cdot 149 + 43\cdot 149^{2} + 6\cdot 149^{3} + 131\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 120 + 117\cdot 149 + 6\cdot 149^{2} + 64\cdot 149^{3} + 55\cdot 149^{4} +O(149^{5})\) 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.