Properties

 Label 1.11_29.2t1.1c1 Dimension 1 Group $C_2$ Conductor $11 \cdot 29$ Root number 1 Frobenius-Schur indicator 1

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Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $319= 11 \cdot 29$ Artin number field: Splitting field of $f= x^{2} - x + 80$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-319}{\bullet}\right)$$

Galois action

Roots of defining polynomial

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

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.