Properties

Label 3.1099.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $1099$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(1099\)\(\medspace = 7 \cdot 157 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.1099.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.1099.2t1.a.a
Projective image: $S_4$
Projective field: Galois closure of 4.2.1099.1

Defining polynomial

$f(x)$$=$$ x^{4} - x^{3} + x^{2} - 3 x + 1 $.

The roots of $f$ are computed in $\Q_{ 317 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 116 + 117\cdot 317 + 120\cdot 317^{2} + 83\cdot 317^{3} + 188\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 168 + 199\cdot 317 + 307\cdot 317^{2} + 80\cdot 317^{3} + 173\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 170 + 64\cdot 317 + 126\cdot 317^{2} + 41\cdot 317^{3} + 56\cdot 317^{4} +O\left(317^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 181 + 252\cdot 317 + 79\cdot 317^{2} + 111\cdot 317^{3} + 216\cdot 317^{4} +O\left(317^{ 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.