# Properties

 Label 1.240.4t1.f Dimension $1$ Group $C_4$ Conductor $240$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$240$$$$\medspace = 2^{4} \cdot 3 \cdot 5$$ Artin number field: Galois closure of 4.0.2304000.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $$1 + 9\cdot 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 6\cdot 13^{4} + 9\cdot 13^{5} +O(13^{6})$$ 1 + 9*13 + 7*13^2 + 5*13^3 + 6*13^4 + 9*13^5+O(13^6) $r_{ 2 }$ $=$ $$2 + 4\cdot 13 + 4\cdot 13^{2} + 13^{3} + 8\cdot 13^{4} + 10\cdot 13^{5} +O(13^{6})$$ 2 + 4*13 + 4*13^2 + 13^3 + 8*13^4 + 10*13^5+O(13^6) $r_{ 3 }$ $=$ $$11 + 8\cdot 13 + 8\cdot 13^{2} + 11\cdot 13^{3} + 4\cdot 13^{4} + 2\cdot 13^{5} +O(13^{6})$$ 11 + 8*13 + 8*13^2 + 11*13^3 + 4*13^4 + 2*13^5+O(13^6) $r_{ 4 }$ $=$ $$12 + 3\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 6\cdot 13^{4} + 3\cdot 13^{5} +O(13^{6})$$ 12 + 3*13 + 5*13^2 + 7*13^3 + 6*13^4 + 3*13^5+O(13^6)

### 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 values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $-1$ $1$ $4$ $(1,3,4,2)$ $\zeta_{4}$ $-\zeta_{4}$ $1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.