Properties

Label 3.17424.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $17424$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(17424\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.17424.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.17424.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 20 + 93\cdot 101 + 62\cdot 101^{2} + 20\cdot 101^{3} + 85\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 73\cdot 101 + 19\cdot 101^{2} + 51\cdot 101^{3} + 17\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 64 + 95\cdot 101 + 15\cdot 101^{2} + 41\cdot 101^{3} + 68\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 95 + 40\cdot 101 + 2\cdot 101^{2} + 89\cdot 101^{3} + 30\cdot 101^{4} +O(101^{5})\) 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.