Properties

Label 2.257.3t2.a.a
Dimension 2
Group $S_3$
Conductor $ 257 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$257 $
Artin number field: Splitting field of $f= x^{3} - x^{2} - 4 x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.257.2t1.a.a

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 27 + 21\cdot 61 + 25\cdot 61^{2} + 29\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 39 + 12\cdot 61 + 61^{2} + 22\cdot 61^{3} + 33\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 57 + 26\cdot 61 + 34\cdot 61^{2} + 9\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 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.