Properties

Label 2.1257.3t2.a.a
Dimension 2
Group $S_3$
Conductor $ 3 \cdot 419 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$1257= 3 \cdot 419 $
Artin number field: Splitting field of 3.3.1257.1 defined by $f= x^{3} - x^{2} - 8 x + 9 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.1257.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.3.1257.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 7 + 9\cdot 13 + 13^{2} + 12\cdot 13^{3} + 7\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 8\cdot 13 + 5\cdot 13^{2} + 8\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 + 7\cdot 13 + 5\cdot 13^{2} + 5\cdot 13^{3} + 8\cdot 13^{4} +O\left(13^{ 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.