Properties

Label 3.1456.4t5.b.a
Dimension $3$
Group $S_4$
Conductor $1456$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(1456\)\(\medspace = 2^{4} \cdot 7 \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.1456.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.91.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.1456.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 8 + 58\cdot 59 + 7\cdot 59^{2} + 3\cdot 59^{3} + 25\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 + 9\cdot 59 + 21\cdot 59^{2} + 54\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 + 53\cdot 59 + 33\cdot 59^{2} + 34\cdot 59^{3} + 8\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 47 + 55\cdot 59 + 54\cdot 59^{2} + 20\cdot 59^{3} + 30\cdot 59^{4} +O(59^{5})\) 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$