Properties

Label 2.229.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 229 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$229 $
Artin number field: Splitting field of $f= x^{3} - 4 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.229.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 21 + 2\cdot 37 + 14\cdot 37^{2} + 35\cdot 37^{3} + 18\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 + 5\cdot 37 + 3\cdot 37^{2} + 5\cdot 37^{3} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 29 + 28\cdot 37 + 19\cdot 37^{2} + 33\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 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.