# Properties

 Label 1.145.4t1.a.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: 4.0.609725.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{145}(104,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 33 x^{2} - 107 x + 139$$  .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$1 + 5\cdot 13 + 13^{2} + 7\cdot 13^{3} + 13^{4} +O(13^{5})$$ $r_{ 2 }$ $=$ $$2 + 10\cdot 13 + 9\cdot 13^{3} + 13^{4} +O(13^{5})$$ $r_{ 3 }$ $=$ $$3 + 4\cdot 13^{2} + 4\cdot 13^{3} + 7\cdot 13^{4} +O(13^{5})$$ $r_{ 4 }$ $=$ $$8 + 10\cdot 13 + 6\cdot 13^{2} + 5\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.