Properties

Label 2.148.3t2.a.a
Dimension $2$
Group $S_3$
Conductor $148$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(148\)\(\medspace = 2^{2} \cdot 37 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 3.3.148.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: even
Determinant: 1.37.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.3.148.1

Defining polynomial

$f(x)$$=$\(x^{3} - x^{2} - 3 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 10 + 6\cdot 67 + 45\cdot 67^{2} + 30\cdot 67^{3} + 50\cdot 67^{4} +O(67^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 62 + 17\cdot 67 + 27\cdot 67^{2} + 39\cdot 67^{3} + 14\cdot 67^{4} +O(67^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 63 + 42\cdot 67 + 61\cdot 67^{2} + 63\cdot 67^{3} + 67^{4} +O(67^{5})\)  Toggle raw display

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.