Properties

Label 1.5_13.2t1.1c1
Dimension 1
Group $C_2$
Conductor $ 5 \cdot 13 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:$65= 5 \cdot 13 $
Artin number field: Splitting field of $f= x^{2} - x - 16 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2$
Parity: Even
Corresponding Dirichlet character: \(\displaystyle\left(\frac{65}{\bullet}\right)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 2 + 3\cdot 7 + 6\cdot 7^{2} + 3\cdot 7^{3} + 3\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 2 } &= 6 + 3\cdot 7 + 3\cdot 7^{3} + 3\cdot 7^{4} +O\left(7^{ 5 }\right) \\ \end{aligned}\]

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

Cycle notation
$(1,2)$

Character values on conjugacy classes

SizeOrderAction 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.