Properties

Label 3.1461681.6t8.a.a
Dimension $3$
Group $S_4$
Conductor $1461681$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(1461681\)\(\medspace = 3^{2} \cdot 13^{2} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.47151.1
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 4.2.47151.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 4x^{2} + 7x + 10 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 34 + 26\cdot 173 + 87\cdot 173^{2} + 43\cdot 173^{3} + 24\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 47 + 105\cdot 173 + 152\cdot 173^{2} + 90\cdot 173^{3} + 75\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 130 + 21\cdot 173 + 50\cdot 173^{2} + 82\cdot 173^{3} + 19\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 136 + 19\cdot 173 + 56\cdot 173^{2} + 129\cdot 173^{3} + 53\cdot 173^{4} +O(173^{5})\) 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.