# Properties

 Label 1.145.4t1.c Dimension $1$ Group $C_4$ Conductor $145$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$145$$$$\medspace = 5 \cdot 29$$ Artin number field: Galois closure of 4.4.3048625.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$2 + 4\cdot 17 + 14\cdot 17^{2} + 11\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$9 + 7\cdot 17 + 12\cdot 17^{2} + 10\cdot 17^{3} +O(17^{5})$$ $r_{ 3 }$ $=$ $$10 + 4\cdot 17 + 13\cdot 17^{2} + 17^{3} + 14\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$14 + 11\cdot 17^{2} + 9\cdot 17^{3} + 10\cdot 17^{4} +O(17^{5})$$

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

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

### Character values on conjugacy classes

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