Properties

Label 2.1099.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $1099$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(1099\)\(\medspace = 7 \cdot 157 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.1099.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.1099.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.1.1099.1

Defining polynomial

$f(x)$$=$$ x^{3} - x^{2} - x - 6 $.

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 13 + 28\cdot 41 + 33\cdot 41^{2} + 36\cdot 41^{3} + 33\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 33 + 6\cdot 41 + 21\cdot 41^{2} + 21\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 + 5\cdot 41 + 27\cdot 41^{2} + 23\cdot 41^{3} + 18\cdot 41^{4} +O\left(41^{ 5 }\right)$

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation
$(1,2,3)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character value
$1$$1$$()$$2$
$3$$2$$(1,2)$$0$
$2$$3$$(1,2,3)$$-1$

The blue line marks the conjugacy class containing complex conjugation.