Properties

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

Related objects

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:\(7\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of \(\Q(\sqrt{-7}) \)
Galois orbit size: $1$
Smallest permutation container: $C_2$
Parity: odd
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

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\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 9\cdot 11 + 9\cdot 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} +O\left(11^{ 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 values
$c1$
$1$ $1$ $()$ $1$
$1$ $2$ $(1,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.