Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut 12 x^{4} \) \(\mathstrut +\mathstrut 13 x^{3} \) \(\mathstrut +\mathstrut 19 x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut -\mathstrut 5 \)
| $\times$ | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 61, ·)\) |
|---|---|---|---|---|---|---|
| \(\chi_{ 65 }(1, ·)\) | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 61, ·)\) |
| \(\chi_{ 65 }(16, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 61, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 1, ·)\) |
| \(\chi_{ 65 }(9, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 61, ·)\) | \(\chi_{ 65 } ( 29, ·)\) |
| \(\chi_{ 65 }(29, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 61, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 14, ·)\) |
| \(\chi_{ 65 }(14, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 61, ·)\) | \(\chi_{ 65 } ( 16, ·)\) | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 9, ·)\) |
| \(\chi_{ 65 }(61, ·)\) | \(\chi_{ 65 } ( 61, ·)\) | \(\chi_{ 65 } ( 1, ·)\) | \(\chi_{ 65 } ( 29, ·)\) | \(\chi_{ 65 } ( 14, ·)\) | \(\chi_{ 65 } ( 9, ·)\) | \(\chi_{ 65 } ( 16, ·)\) |