# Properties

 Label 1.80.4t1.a.b Dimension $1$ Group $C_4$ Conductor $80$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$80$$$$\medspace = 2^{4} \cdot 5$$ Artin number field: Galois closure of 4.4.51200.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{80}(29,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{4} - 20 x^{2} + 50$.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $2 + 2\cdot 7 + 5\cdot 7^{2} + 2\cdot 7^{3} + 5\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 2 }$ $=$ $3 + 4\cdot 7^{2} + 5\cdot 7^{4} + 4\cdot 7^{5} + 2\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 3 }$ $=$ $4 + 6\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} + 2\cdot 7^{5} + 4\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 4 }$ $=$ $5 + 4\cdot 7 + 7^{2} + 4\cdot 7^{3} + 6\cdot 7^{4} + 6\cdot 7^{5} + 7^{6} +O\left(7^{ 7 }\right)$

## 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 value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $1$ $4$ $(1,3,4,2)$ $\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.