Properties

Label 3.5_353.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 5 \cdot 353 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1765= 5 \cdot 353 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - x^{2} - x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.5_353.2t1.1c1

Galois action

Roots of defining polynomial

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\left(173^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 52 + 100\cdot 173 + 46\cdot 173^{2} + 123\cdot 173^{3} + 162\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 131 + 120\cdot 173 + 45\cdot 173^{2} + 56\cdot 173^{3} + 155\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 150 + 106\cdot 173 + 54\cdot 173^{2} + 171\cdot 173^{3} + 141\cdot 173^{4} +O\left(173^{ 5 }\right)$

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.