Properties

Label 2.1083.3t2.b.a
Dimension $2$
Group $S_3$
Conductor $1083$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(1083\)\(\medspace = 3 \cdot 19^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.1083.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.1.1083.1

Defining polynomial

$f(x)$$=$$ x^{3} - x^{2} - 6 x - 12 $.

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 8 + 81\cdot 97 + 33\cdot 97^{2} + 85\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 89\cdot 97 + 3\cdot 97^{2} + 86\cdot 97^{3} + 77\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 81 + 23\cdot 97 + 59\cdot 97^{2} + 22\cdot 97^{3} + 39\cdot 97^{4} +O\left(97^{ 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.