Properties

 Label 1.105.4t1.a Dimension 1 Group $C_4$ Conductor $3 \cdot 5 \cdot 7$ Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $105= 3 \cdot 5 \cdot 7$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 26 x^{2} - 26 x + 151$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Projective image: $C_1$ Projective field: $$\Q$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 9\cdot 19^{2} + 7\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 + 14\cdot 19 + 18\cdot 19^{3} + 12\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 + 16\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 + 6\cdot 19 + 9\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,3)(2,4)$ $-1$ $-1$ $1$ $4$ $(1,2,3,4)$ $\zeta_{4}$ $-\zeta_{4}$ $1$ $4$ $(1,4,3,2)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.