Properties

Label 1.17.2t1.a.a
Dimension $1$
Group $C_2$
Conductor $17$
Root number $1$
Indicator $1$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_2$
Conductor: \(17\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of \(\Q(\sqrt{17}) \)
Galois orbit size: $1$
Smallest permutation container: $C_2$
Parity: even
Dirichlet character: \(\displaystyle\left(\frac{17}{\bullet}\right)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{2} - x - 4 $.

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 6 + 13 + 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 + 11\cdot 13 + 11\cdot 13^{2} + 4\cdot 13^{3} +O\left(13^{ 5 }\right)$

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.