# Properties

 Label 1.60.4t1.a.a Dimension 1 Group $C_4$ Conductor $2^{2} \cdot 3 \cdot 5$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $60= 2^{2} \cdot 3 \cdot 5$ Artin number field: Splitting field of 4.0.18000.1 defined by $f= x^{4} + 15 x^{2} + 45$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{60}(23,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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

### 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.