Basic invariants
Dimension: | $1$ |
Group: | $C_2$ |
Conductor: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of \(\Q(\sqrt{-13}) \) |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2$ |
Parity: | odd |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 6\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} +O(7^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 + 4\cdot 7^{2} + 5\cdot 7^{4} +O(7^{5})\) |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)$ | $-1$ |