Properties

Label 2.2e2_37.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 2^{2} \cdot 37 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$148= 2^{2} \cdot 37 $
Artin number field: Splitting field of $f= x^{3} - x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 10 + 6\cdot 67 + 45\cdot 67^{2} + 30\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 62 + 17\cdot 67 + 27\cdot 67^{2} + 39\cdot 67^{3} + 14\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 63 + 42\cdot 67 + 61\cdot 67^{2} + 63\cdot 67^{3} + 67^{4} +O\left(67^{ 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.