Properties

 Label 1.7.2t1.a.a Dimension 1 Group $C_2$ Conductor $7$ Root number 1 Frobenius-Schur indicator 1

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Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $7$ Artin number field: Splitting field of $$\Q(\sqrt{-7})$$ defined by $f= x^{2} - x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-7}{\bullet}\right)$$ Projective image: $C_1$ Projective field: $$\Q$$

Galois action

Roots of defining polynomial

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

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.