Properties

Label 2.17_41.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 17 \cdot 41 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$697= 17 \cdot 41 $
Artin number field: Splitting field of $f= x^{3} - 7 x - 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.17_41.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 1 + 8\cdot 11 + 6\cdot 11^{2} + 3\cdot 11^{3} + 9\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 2 } &= 2 + 9\cdot 11 + 11^{2} + 4\cdot 11^{3} + 8\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 3 } &= 8 + 4\cdot 11 + 2\cdot 11^{2} + 3\cdot 11^{3} + 4\cdot 11^{4} +O\left(11^{ 5 }\right) \\ \end{aligned}\]

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.