Properties

 Label 1.5_7.2t1.1c1 Dimension 1 Group $C_2$ Conductor $5 \cdot 7$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $35= 5 \cdot 7$ Artin number field: Splitting field of $f= x^{2} - x + 9$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-35}{\bullet}\right)$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 3 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 3^{2} + 3^{4} +O\left(3^{ 5 }\right) \\ r_{ 2 } &= 1 + 2\cdot 3^{2} + 2\cdot 3^{3} + 3^{4} +O\left(3^{ 5 }\right) \\ \end{aligned}

Generators of the action on the roots $r_{ 1 }, r_{ 2 }$

 Cycle notation $(1,2)$

Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.