# Properties

 Label 1.35.4t1.a.b Dimension $1$ Group $C_4$ Conductor $35$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$35$$$$\medspace = 5 \cdot 7$$ Artin field: 4.4.6125.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{35}(13,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 9 x^{2} + 9 x + 11$$  .

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$6\cdot 11 + 4\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$1 + 7\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$3 + 7\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{4} +O(11^{5})$$ $r_{ 4 }$ $=$ $$8 + 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})$$

## 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 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.