Properties

Label 2.676.3t2.b.a
Dimension $2$
Group $S_3$
Conductor $676$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(676\)\(\medspace = 2^{2} \cdot 13^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 3.1.676.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.1.676.1

Defining polynomial

$f(x)$$=$$ x^{3} - x^{2} - 4 x + 12 $.

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 58 + 85\cdot 89 + 36\cdot 89^{2} + 27\cdot 89^{3} + 84\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 59 + 73\cdot 89 + 49\cdot 89^{2} + 16\cdot 89^{3} + 70\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 62 + 18\cdot 89 + 2\cdot 89^{2} + 45\cdot 89^{3} + 23\cdot 89^{4} +O\left(89^{ 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.