Properties

Label 1.53.2t1.a.a
Dimension $1$
Group $C_2$
Conductor $53$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $1$
Group: $C_2$
Conductor: \(53\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of \(\Q(\sqrt{53}) \)
Galois orbit size: $1$
Smallest permutation container: $C_2$
Parity: even
Dirichlet character: \(\displaystyle\left(\frac{53}{\bullet}\right)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{2} - x - 13 $.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 3 + 3\cdot 7 + 2\cdot 7^{2} + 4\cdot 7^{3} + 6\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 + 3\cdot 7 + 4\cdot 7^{2} + 2\cdot 7^{3} +O\left(7^{ 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.