Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \)
| $\times$ | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 12, ·)\) |
|---|---|---|---|---|---|---|
| \(\chi_{ 13 }(1, ·)\) | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 12, ·)\) |
| \(\chi_{ 13 }(3, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 12, ·)\) | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 10, ·)\) |
| \(\chi_{ 13 }(4, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 12, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 9, ·)\) |
| \(\chi_{ 13 }(9, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 12, ·)\) | \(\chi_{ 13 } ( 4, ·)\) |
| \(\chi_{ 13 }(10, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 1, ·)\) | \(\chi_{ 13 } ( 12, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 3, ·)\) |
| \(\chi_{ 13 }(12, ·)\) | \(\chi_{ 13 } ( 12, ·)\) | \(\chi_{ 13 } ( 10, ·)\) | \(\chi_{ 13 } ( 9, ·)\) | \(\chi_{ 13 } ( 4, ·)\) | \(\chi_{ 13 } ( 3, ·)\) | \(\chi_{ 13 } ( 1, ·)\) |