Properties

 Label 3.331e2.6t8.3c1 Dimension 3 Group $S_4$ Conductor $331^{2}$ Root number 1 Frobenius-Schur indicator 1

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Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $109561= 331^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} + x - 1$ 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 $\Q_{ 113 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 40 + 35\cdot 113 + 41\cdot 113^{2} + 79\cdot 113^{3} + 67\cdot 113^{4} +O\left(113^{ 5 }\right) \\ r_{ 2 } &= 94 + 58\cdot 113 + 20\cdot 113^{2} + 65\cdot 113^{3} + 55\cdot 113^{4} +O\left(113^{ 5 }\right) \\ r_{ 3 } &= 96 + 64\cdot 113 + 113^{2} + 99\cdot 113^{3} + 91\cdot 113^{4} +O\left(113^{ 5 }\right) \\ r_{ 4 } &= 110 + 66\cdot 113 + 49\cdot 113^{2} + 95\cdot 113^{3} + 10\cdot 113^{4} +O\left(113^{ 5 }\right) \\ \end{aligned}

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.