Properties

Label 3.1765.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $1765$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(1765\)\(\medspace = 5 \cdot 353 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.1765.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1765.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.0.1765.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 14 + 18\cdot 173 + 26\cdot 173^{2} + 168\cdot 173^{3} + 58\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 52 + 100\cdot 173 + 46\cdot 173^{2} + 123\cdot 173^{3} + 162\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 131 + 120\cdot 173 + 45\cdot 173^{2} + 56\cdot 173^{3} + 155\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 150 + 106\cdot 173 + 54\cdot 173^{2} + 171\cdot 173^{3} + 141\cdot 173^{4} +O(173^{5})\) 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 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$