Properties

 Label 3.3e2_2351e2.6t8.1c1 Dimension 3 Group $S_4$ Conductor $3^{2} \cdot 2351^{2}$ Root number 1 Frobenius-Schur indicator 1

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Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $49744809= 3^{2} \cdot 2351^{2}$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} - 4 x^{2} + 3 x + 3$ 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 $\Q_{ 229 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 190\cdot 229 + 70\cdot 229^{2} + 68\cdot 229^{3} + 161\cdot 229^{4} +O\left(229^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 132\cdot 229 + 214\cdot 229^{2} + 200\cdot 229^{3} + 51\cdot 229^{4} +O\left(229^{ 5 }\right)$ $r_{ 3 }$ $=$ $77 + 222\cdot 229 + 157\cdot 229^{2} + 103\cdot 229^{3} + 160\cdot 229^{4} +O\left(229^{ 5 }\right)$ $r_{ 4 }$ $=$ $128 + 142\cdot 229 + 14\cdot 229^{2} + 85\cdot 229^{3} + 84\cdot 229^{4} +O\left(229^{ 5 }\right)$

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.