Properties

 Label 3.3e2_7e2_19e2.4t4.1c1 Dimension 3 Group $A_4$ Conductor $3^{2} \cdot 7^{2} \cdot 19^{2}$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $159201= 3^{2} \cdot 7^{2} \cdot 19^{2}$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 7 x^{2} + 21 x + 63$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_4$ Parity: Even Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 4\cdot 47 + 47^{2} + 38\cdot 47^{3} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $18 + 21\cdot 47 + 3\cdot 47^{2} + 26\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $31 + 31\cdot 47 + 31\cdot 47^{2} + 10\cdot 47^{3} + 19\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $38 + 36\cdot 47 + 10\cdot 47^{2} + 19\cdot 47^{3} + 5\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

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