Properties

Label 2.940.3t2.b.a
Dimension $2$
Group $S_3$
Conductor $940$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $S_3$
Conductor: \(940\)\(\medspace = 2^{2} \cdot 5 \cdot 47 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 3.3.940.1
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: even
Determinant: 1.940.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.3.940.1

Defining polynomial

$f(x)$$=$ \( x^{3} - 7x - 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 46 + 88\cdot 89 + 48\cdot 89^{2} + 7\cdot 89^{3} + 64\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 49 + 74\cdot 89 + 7\cdot 89^{2} + 34\cdot 89^{3} + 26\cdot 89^{4} +O(89^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 83 + 14\cdot 89 + 32\cdot 89^{2} + 47\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\) Copy content 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.