Properties

Label 2.5_269.3t2.1c1
Dimension 2
Group $S_3$
Conductor $ 5 \cdot 269 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:$1345= 5 \cdot 269 $
Artin number field: Splitting field of $f= x^{3} - 7 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_3$
Parity: Even
Determinant: 1.5_269.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 1 + 5\cdot 7 + 4\cdot 7^{2} + 2\cdot 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 2 } &= 2 + 3\cdot 7 + 6\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 3 } &= 4 + 5\cdot 7 + 7^{2} + 4\cdot 7^{3} + 3\cdot 7^{4} +O\left(7^{ 5 }\right) \\ \end{aligned}\]

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.