Properties

 Label 2.2e2_3_127.3t2.1c1 Dimension 2 Group $S_3$ Conductor $2^{2} \cdot 3 \cdot 127$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $1524= 2^{2} \cdot 3 \cdot 127$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 7 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.3_127.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 12 + 8\cdot 19 + 13\cdot 19^{3} + 6\cdot 19^{4} +O\left(19^{ 5 }\right) \\ r_{ 2 } &= 13 + 7\cdot 19 + 9\cdot 19^{2} + 10\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right) \\ r_{ 3 } &= 14 + 2\cdot 19 + 9\cdot 19^{2} + 14\cdot 19^{3} + 13\cdot 19^{4} +O\left(19^{ 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

 Size Order Action 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.