# Properties

 Label 1.145.4t1.b.a Dimension $1$ Group $C_4$ Conductor $145$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$145$$$$\medspace = 5 \cdot 29$$ Artin field: 4.0.105125.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{145}(57,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 36 x^{2} - 36 x + 281$$  .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$2 + 4\cdot 19^{2} + 13\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$3 + 2\cdot 19 + 6\cdot 19^{2} + 19^{3} + 14\cdot 19^{4} +O(19^{5})$$ $r_{ 3 }$ $=$ $$5 + 2\cdot 19 + 11\cdot 19^{2} + 12\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$10 + 14\cdot 19 + 16\cdot 19^{2} + 10\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)(3,4)$ $-1$ $1$ $4$ $(1,4,2,3)$ $\zeta_{4}$ $1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.