Properties

 Label 3.11e2_233e2.6t8.1c1 Dimension 3 Group $S_4$ Conductor $11^{2} \cdot 233^{2}$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $6568969= 11^{2} \cdot 233^{2}$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 4 x^{2} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $x^{2} + 6 x + 3$
Roots:
 $r_{ 1 }$ $=$ $3 a + 1 + \left(3 a + 2\right)\cdot 7 + \left(6 a + 5\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(3 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 2\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 6\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a + 4 + \left(3 a + 2\right)\cdot 7 + 7^{2} + \left(4 a + 2\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 + 6\cdot 7 + 4\cdot 7^{2} + 6\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2,3,4)$ $(1,2)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $-1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $1$
The blue line marks the conjugacy class containing complex conjugation.