Properties

Label 1.7.2t1.a.a
Dimension $1$
Group $C_2$
Conductor $7$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $1$
Group: $C_2$
Conductor: \(7\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin field: \(\Q(\sqrt{-7}) \)
Galois orbit size: $1$
Smallest permutation container: $C_2$
Parity: odd
Dirichlet character: \(\displaystyle\left(\frac{-7}{\bullet}\right)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{2} - x + 2\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 11 + 11^{2} + 7\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 9\cdot 11 + 9\cdot 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)  Toggle raw display

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.